1.4 Properties of whole numbers

Achievable Praxis Core: Math (5733)

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Properties of whole numbers

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This section covers the core building blocks of number theory: whole numbers, factors, multiples, prime numbers, and divisibility rules. These ideas show up often on the Praxis Core exam and support skills like simplifying fractions, solving ratio problems, spotting patterns, and working with algebraic expressions. When you know how numbers are built, you can break them apart efficiently and use quick rules to speed up computation.

Definitions

Whole numbers: Whole numbers include $0$ and all positive integers: $0, 1, 2, 3, 4, 5, \dots$ They do not include negative numbers, fractions, or decimals.

Whole numbers are the starting point for many math topics. You use them for counting, labeling positions, and describing steps in a process. In this section, you’ll connect whole numbers to factorization, divisibility, and prime structure so you can see how numbers are constructed and how they relate to each other.

Factors

Factors tell you which smaller whole numbers multiply to make a larger number. Factors matter any time you simplify fractions, compare ratios, or break a number into primes. When you find factors, you’re uncovering the “parts” a number can be split into.

Definitions

Factor: A factor of a whole number is a number that divides it evenly, with no remainder.

Listing factors

Before using tools like prime factorization, it helps to practice listing factors for smaller numbers.

Example: Listing factors The number $12$ has the following factors: $1, 2, 3, 4, 6, 12$ because each divides $12$ exactly.

$12 \div 1 = 12$

$12 \div 2 = 6$

$12 \div 3 = 4$

$12 \div 4 = 3$

$12 \div 6 = 2$

$12 \div 12 = 1$ Factors always come in pairs, one below the square root and one above.

Answer: $1, 2, 3, 4, 6, 12$

Factor pairs help you work efficiently. Once you’ve checked all possible factors up to the square root, you’ve automatically found the larger factors as the matching partners.

Divisibility rules

Divisibility rules let you check whether one number divides another without doing long division. They’re based on digit patterns, so they’re fast and help reduce arithmetic mistakes.

Useful rules

Divisible by 2: Last digit is $0, 2, 4, 6, 8$
Divisible by 3: Sum of digits divisible by $3$
Divisible by 4: Last two digits form a number divisible by $4$
Divisible by 5: Last digit is $0$ or $5$
Divisible by 6: Divisible by both $2$ and $3$
Divisible by 9: Sum of digits divisible by $9$
Divisible by 10: Last digit is $0$

Example: Using a divisibility rule Is $132$ divisible by $3$ ?

$1 + 3 + 2 = 6$

$6$ is divisible by $3$

Answer: Yes

These rules are especially useful when you’re simplifying fractions or breaking numbers into prime factors.

Using divisibility to find factors

Divisibility rules give you a systematic way to factor numbers: test small primes, divide, and keep going until what’s left is prime.

Example: Factoring through divisibility

Example: Find factors of $36$ We start with small prime numbers. Because $36$ is even, it is divisible by $2$ .

$36 \div 2 = 18$ This shows that $2$ and $18$ are a factor pair of $36$ .

$18 \div 2 = 9$

Dividing again by $2$ shows that $4 = 2 \times 2$ is a factor of $36$ , with $9$ remaining.

At this point, $9$ is no longer divisible by $2$ , so we switch to the next smallest prime number, $3$ .

$9 \div 3 = 3$

$3 \div 3 = 1$

This confirms that $9 = 3 \times 3$ . Combining all the prime factors used:

Prime factorization: $36 = 2^{2} \times 3^{2}$

The full list of factors comes from pairing these primes in every possible way:

Answer: Factors: $1, 2, 3, 4, 6, 9, 12, 18, 36$

Example: Find factors of $72$ We again start with the smallest prime number. Since $72$ is even, we repeatedly divide by $2$ .

$72 \div 2 = 36$

$36 \div 2 = 18$

$18 \div 2 = 9$

These steps show that $72$ contains three factors of $2$ , since $72 = 2 \times 2 \times 2 \times 9$ .

Because $9$ is no longer divisible by $2$ , we switch to $3$ .

$9 \div 3 = 3$

$3 \div 3 = 1$

This shows that $9 = 3 \times 3$ . Putting everything together:

Prime factorization: $72 = 2^{3} \times 3^{2}$

To find all factors, we form products using different combinations of the prime factors. For example:

$2 \times 2 = 4$

$2 \times 2 \times 2 = 8$

$3 \times 4 = 12$

$3 \times 6 = 18$

$3^{2} \times 2^{3} = 72$

Answer: Factors: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72$

Prime factorization shows the “building blocks” of a number, but it doesn’t automatically list every factor. To list all factors, you still have to combine the prime factors in all possible ways. That’s why a number with a short prime factorization can still have many total factors.

Factor trees

Factor trees give you a visual way to break a number into primes. Each branch splits a number into smaller factors until only prime numbers remain.

Definitions

Factor tree: A diagram that splits a number into factors until only prime numbers remain.

Example: Factor tree for $36$ Prime factorization: $36 = 2^{2} \times 3^{2}$

Example: Factor tree for $60$

(spoiler)

Prime factorization: $60 = 2^{2} \times 3 \times 5$

Full factor list: $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$

Factor tree for 60

Common factors, GCF, and GCD

Common factors are factors shared by two or more numbers. The greatest common factor (GCF) is the largest shared factor, which is why it’s so useful for simplifying fractions and reducing ratios. The term greatest common divisor (GCD) means the same thing as GCF - “factor” and “divisor” are two names for a number that divides another number evenly.

Definitions

Common factor: A factor shared by two or more numbers
Greatest common factor (GCF) / Greatest common divisor (GCD): The largest whole number that divides each of the given numbers evenly

A reliable method for finding the GCF or GCD:

Write prime factorizations of each number.
Identify the primes they have in common.
Use the smallest exponent of each shared prime.
Multiply these together.

Example: GCF of $48$ and $180$

(spoiler)

$48 = 2^{4} \times 3$

$180 = 2^{2} \times 3^{2} \times 5$

Shared lowest powers: $2^{2} \times 3 = 12$

Answer: $12$

Example: GCD of $252$ and $336$

(spoiler)

$252 = 2^{2} \times 3^{2} \times 7$

$336 = 2^{4} \times 3 \times 7$

GCD = $2^{2} \times 3 \times 7 = 84$

Answer: $84$

Multiples and the least common multiple (LCM)

Multiples “expand outward” from a number by repeated multiplication. The LCM is the smallest number that both values divide into evenly.

Definitions

Multiple: A number of the form $n \times k$
Common multiple: A number that is a multiple of two or more values
Least common multiple (LCM): The smallest positive common multiple

Example: LCM of $8$ and $18$

$8 = 2^{3}$

$18 = 2 \times 3^{2}$

LCM uses highest powers:

$2^{3} \times 3^{2} = 72$

Answer: $72$

Even and odd numbers

Even and odd numbers describe parity, which shows up in patterns, divisibility, and algebra.

Even: divisible by $2$ ; form $2 k$
Odd: not divisible by $2$ ; form $2 k + 1$

These definitions explain many common patterns. For example, adding two odd numbers always gives an even result, and multiplying an even number by any whole number always gives an even product.

Prime numbers

Prime numbers are the basic “atoms” of number theory. Every whole number greater than $1$ is either prime or composite.

Definitions

Prime number: A number greater than $1$ with exactly two factors: $1$ and itself

Important notes:

$2$ is the only even prime
$1$ is not prime
Composite numbers have additional factors

Testing for primality

To test whether a number is prime, check divisibility using primes up to $n$ .

Sidenote

How factor pairs work with square roots

Every factor comes in a pair: if $d$ divides $N$ , then $N / d$ is the partner. One factor in each pair must be less than or equal to $N$ . Once you pass $N$ , all larger factors already appeared as the “other half” of a smaller factor.

Example: Is $37$ prime?

$37 \approx 6.08$

Test primes $2, 3, 5$

Answer: $37$ is prime

Example: Is $45$ prime?

(spoiler)

$45 \approx 6.7$

Divisible by $3$ and $5$

Answer: Not prime

More practice with divisibility and factorization

Example: Test $234$ for divisibility

(spoiler)

We check divisibility using standard rules for each number.

$2$ : The number ends in $4$ , which is even, so $234$ is divisible by $2$ .
$3$ : Add the digits: $2 + 3 + 4 = 9$ . Since $9$ is divisible by $3$ , $234$ is divisible by $3$ .
$4$ : Look at the last two digits: $34$ . Because $34$ is not divisible by $4$ , $234$ is not divisible by $4$ .
$5$ : The number does not end in $0$ or $5$ , so it is not divisible by $5$ .
$6$ : A number is divisible by $6$ if it is divisible by both $2$ and $3$ . Since both conditions are met, $234$ is divisible by $6$ .
$9$ : The digit sum is $9$ , which is divisible by $9$ , so $234$ is divisible by $9$ .
$10$ : The number does not end in $0$ , so it is not divisible by $10$ .

Answer: Divisible by $2, 3, 6,$ and $9$

Example: Prime factorization of $96$

(spoiler)

Because $96$ is even, we repeatedly divide by $2$ until no longer possible.

$96 \div 2 = 48$
$48 \div 2 = 24$
$24 \div 2 = 12$
$12 \div 2 = 6$
$6 \div 2 = 3$
At this point, $3$ is prime and cannot be divided further.
This shows that $96$ contains five factors of $2$ and one factor of $3$ .

Answer: $96 = 2^{5} \times 3$

Example: Is $91$ prime?

(spoiler)

To determine whether $91$ is prime, we test divisibility by small prime numbers.

$91$ is not even, so it is not divisible by $2$ .
The digit sum is $9 + 1 = 10$ , so it is not divisible by $3$ .
It does not end in $0$ or $5$ , so it is not divisible by $5$ .
Try $7$ : $91 \div 7 = 13$
Since $91$ can be written as a product of two integers greater than $1$ , it is not prime.

Answer: Not prime

Sequences

A sequence is an ordered list of numbers written in a specific order. Each number in the list is called a term, and its position in the list matters. Many real world patterns, such as savings growth, repeated doubling, or weekly increases, can be described using sequences.

Definitions

Sequence: An ordered list of numbers where each term has a specific position
Term: A number in a sequence
Index: The position number of a term in a sequence
Notation: $a_{n}$ means the $n$ -th term of the sequence $a_{1}$ is the first term, $a_{2}$ is the second term, and $a_{n}$ is the $n$ -th term

A common goal is to find a specific term, such as $a_{10}$ , or to write a rule that generates every term in the sequence.

Arithmetic sequences

An arithmetic sequence changes by adding the same constant amount each time. This constant is called the common difference and is usually written as $d$ .

Definitions

Arithmetic sequence: A sequence where the difference between consecutive terms is constant
Common difference: The constant amount added each step, written as $d$

The explicit formula for an arithmetic sequence is:

$a_{n} = a_{1} + (n - 1) d$

Example: Arithmetic sequence Sequence: $4, 9, 14, 19, \dots$

$a_{1} = 4$

$d = 5$

Example: Find the $10$ -th term

(spoiler)

$a_{10} = 4 + (10 - 1) \cdot 5 = 49$

Answer: $49$

Geometric sequences

A geometric sequence changes by multiplying by the same constant each time. This constant is called the common ratio and is usually written as $r$ .

Definitions

Geometric sequence: A sequence where the ratio between consecutive terms is constant
Common ratio: The constant multiplier each step, written as $r$

The explicit formula for a geometric sequence is:

$a_{n} = a_{1} \cdot r^{n - 1}$

Example: Sequence $5, 10, 20, 40, \dots$

$a_{1} = 5$

$r = 2$

Recursive sequences

A recursive sequence is defined by giving the starting term (or terms) and a rule that tells how to find each new term from earlier terms. Instead of jumping directly to $a_{n}$ , a recursive rule generates the sequence one step at a time.

Definitions

Recursive sequence: A sequence defined by one or more starting terms and a rule that uses earlier terms to produce later terms

A recursive definition includes:

An initial term (such as $a_{1}$ )
A recursive rule that uses previous terms to produce the next one

Recursive formulas emphasize process. To find $a_{5}$ , you compute $a_{2}$ , then $a_{3}$ , then $a_{4}$ , and then $a_{5}$ .

Example: Recursive arithmetic sequence

$a_{1} = 3$

$a_{n + 1} = a_{n} + 4$

Working out the first several terms:

$a_{1} = 3$

$a_{2} = a_{1} + 4 = 3 + 4 = 7$

$a_{3} = a_{2} + 4 = 7 + 4 = 11$

$a_{4} = a_{3} + 4 = 11 + 4 = 15$

$a_{5} = a_{4} + 4 = 15 + 4 = 19$

Answer: The sequence becomes: $3, 7, 11, 15, 19, \dots$

Example: Recursive geometric sequence

$a_{1} = 5$

$a_{n + 1} = 2 a_{n}$

Working out the first several terms:

$a_{1} = 5$

$a_{2} = 2 a_{1} = 2 \cdot 5 = 10$

$a_{3} = 2 a_{2} = 2 \cdot 10 = 20$

$a_{4} = 2 a_{3} = 2 \cdot 20 = 40$

$a_{5} = 2 a_{4} = 2 \cdot 40 = 80$

Answer: The sequence becomes: $5, 10, 20, 40, 80, \dots$

Factors divide numbers evenly, and prime factorization reveals the building blocks of whole numbers.
The greatest common factor (GCF) or greatest common divisor (GCD) is the largest shared divisor and is essential for simplifying fractions and ratios.
The least common multiple (LCM) identifies the smallest shared multiple and is important for matching denominators.
Parity (even or odd) and prime structure help predict how numbers behave.
Sequences describe numerical patterns: arithmetic uses repeated addition, geometric uses repeated multiplication, and recursive formulas show how each term follows from the previous one.

Properties of whole numbers

Definitions

Whole numbers: Whole numbers include $0$ and all positive integers: $0, 1, 2, 3, 4, 5, \dots$ They do not include negative numbers, fractions, or decimals.

Factors

Definitions

Factor: A factor of a whole number is a number that divides it evenly, with no remainder.

Listing factors

Before using tools like prime factorization, it helps to practice listing factors for smaller numbers.

Example: Listing factors The number $12$ has the following factors: $1, 2, 3, 4, 6, 12$ because each divides $12$ exactly.

$12 \div 1 = 12$

$12 \div 2 = 6$

$12 \div 3 = 4$

$12 \div 4 = 3$

$12 \div 6 = 2$

$12 \div 12 = 1$ Factors always come in pairs, one below the square root and one above.

Answer: $1, 2, 3, 4, 6, 12$

Factor pairs help you work efficiently. Once you’ve checked all possible factors up to the square root, you’ve automatically found the larger factors as the matching partners.

Divisibility rules

Divisibility rules let you check whether one number divides another without doing long division. They’re based on digit patterns, so they’re fast and help reduce arithmetic mistakes.

Useful rules

Divisible by 2: Last digit is $0, 2, 4, 6, 8$
Divisible by 3: Sum of digits divisible by $3$
Divisible by 4: Last two digits form a number divisible by $4$
Divisible by 5: Last digit is $0$ or $5$
Divisible by 6: Divisible by both $2$ and $3$
Divisible by 9: Sum of digits divisible by $9$
Divisible by 10: Last digit is $0$

Example: Using a divisibility rule Is $132$ divisible by $3$ ?

$1 + 3 + 2 = 6$

$6$ is divisible by $3$

Answer: Yes

These rules are especially useful when you’re simplifying fractions or breaking numbers into prime factors.

Using divisibility to find factors

Divisibility rules give you a systematic way to factor numbers: test small primes, divide, and keep going until what’s left is prime.

Example: Factoring through divisibility

Example: Find factors of $36$ We start with small prime numbers. Because $36$ is even, it is divisible by $2$ .

$36 \div 2 = 18$ This shows that $2$ and $18$ are a factor pair of $36$ .

$18 \div 2 = 9$

Dividing again by $2$ shows that $4 = 2 \times 2$ is a factor of $36$ , with $9$ remaining.

At this point, $9$ is no longer divisible by $2$ , so we switch to the next smallest prime number, $3$ .

$9 \div 3 = 3$

$3 \div 3 = 1$

This confirms that $9 = 3 \times 3$ . Combining all the prime factors used:

Prime factorization: $36 = 2^{2} \times 3^{2}$

The full list of factors comes from pairing these primes in every possible way:

Answer: Factors: $1, 2, 3, 4, 6, 9, 12, 18, 36$

Example: Find factors of $72$ We again start with the smallest prime number. Since $72$ is even, we repeatedly divide by $2$ .

$72 \div 2 = 36$

$36 \div 2 = 18$

$18 \div 2 = 9$

These steps show that $72$ contains three factors of $2$ , since $72 = 2 \times 2 \times 2 \times 9$ .

Because $9$ is no longer divisible by $2$ , we switch to $3$ .

$9 \div 3 = 3$

$3 \div 3 = 1$

This shows that $9 = 3 \times 3$ . Putting everything together:

Prime factorization: $72 = 2^{3} \times 3^{2}$

To find all factors, we form products using different combinations of the prime factors. For example:

$2 \times 2 = 4$

$2 \times 2 \times 2 = 8$

$3 \times 4 = 12$

$3 \times 6 = 18$

$3^{2} \times 2^{3} = 72$

Answer: Factors: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72$

Factor trees

Factor trees give you a visual way to break a number into primes. Each branch splits a number into smaller factors until only prime numbers remain.

Definitions

Factor tree: A diagram that splits a number into factors until only prime numbers remain.

Example: Factor tree for $36$ Prime factorization: $36 = 2^{2} \times 3^{2}$

Example: Factor tree for $60$

(spoiler)

Prime factorization: $60 = 2^{2} \times 3 \times 5$

Full factor list: $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$

Common factors, GCF, and GCD

Definitions

Common factor: A factor shared by two or more numbers
Greatest common factor (GCF) / Greatest common divisor (GCD): The largest whole number that divides each of the given numbers evenly

A reliable method for finding the GCF or GCD:

Write prime factorizations of each number.
Identify the primes they have in common.
Use the smallest exponent of each shared prime.
Multiply these together.

Example: GCF of $48$ and $180$

(spoiler)

$48 = 2^{4} \times 3$

$180 = 2^{2} \times 3^{2} \times 5$

Shared lowest powers: $2^{2} \times 3 = 12$

Answer: $12$

Example: GCD of $252$ and $336$

(spoiler)

$252 = 2^{2} \times 3^{2} \times 7$

$336 = 2^{4} \times 3 \times 7$

GCD = $2^{2} \times 3 \times 7 = 84$

Answer: $84$

Multiples and the least common multiple (LCM)

Multiples “expand outward” from a number by repeated multiplication. The LCM is the smallest number that both values divide into evenly.

Definitions

Multiple: A number of the form $n \times k$
Common multiple: A number that is a multiple of two or more values
Least common multiple (LCM): The smallest positive common multiple

Example: LCM of $8$ and $18$

$8 = 2^{3}$

$18 = 2 \times 3^{2}$

LCM uses highest powers:

$2^{3} \times 3^{2} = 72$

Answer: $72$

Even and odd numbers

Even and odd numbers describe parity, which shows up in patterns, divisibility, and algebra.

Even: divisible by $2$ ; form $2 k$
Odd: not divisible by $2$ ; form $2 k + 1$

These definitions explain many common patterns. For example, adding two odd numbers always gives an even result, and multiplying an even number by any whole number always gives an even product.

Prime numbers

Prime numbers are the basic “atoms” of number theory. Every whole number greater than $1$ is either prime or composite.

Definitions

Prime number: A number greater than $1$ with exactly two factors: $1$ and itself

Important notes:

$2$ is the only even prime
$1$ is not prime
Composite numbers have additional factors

Testing for primality

To test whether a number is prime, check divisibility using primes up to $n$ .

Sidenote

How factor pairs work with square roots

Example: Is $37$ prime?

$37 \approx 6.08$

Test primes $2, 3, 5$

Answer: $37$ is prime

Example: Is $45$ prime?

(spoiler)

$45 \approx 6.7$

Divisible by $3$ and $5$

Answer: Not prime

More practice with divisibility and factorization

Example: Test $234$ for divisibility

(spoiler)

We check divisibility using standard rules for each number.

$2$ : The number ends in $4$ , which is even, so $234$ is divisible by $2$ .
$3$ : Add the digits: $2 + 3 + 4 = 9$ . Since $9$ is divisible by $3$ , $234$ is divisible by $3$ .
$4$ : Look at the last two digits: $34$ . Because $34$ is not divisible by $4$ , $234$ is not divisible by $4$ .
$5$ : The number does not end in $0$ or $5$ , so it is not divisible by $5$ .
$6$ : A number is divisible by $6$ if it is divisible by both $2$ and $3$ . Since both conditions are met, $234$ is divisible by $6$ .
$9$ : The digit sum is $9$ , which is divisible by $9$ , so $234$ is divisible by $9$ .
$10$ : The number does not end in $0$ , so it is not divisible by $10$ .

Answer: Divisible by $2, 3, 6,$ and $9$

Example: Prime factorization of $96$

(spoiler)

Because $96$ is even, we repeatedly divide by $2$ until no longer possible.

$96 \div 2 = 48$
$48 \div 2 = 24$
$24 \div 2 = 12$
$12 \div 2 = 6$
$6 \div 2 = 3$
At this point, $3$ is prime and cannot be divided further.
This shows that $96$ contains five factors of $2$ and one factor of $3$ .

Answer: $96 = 2^{5} \times 3$

Example: Is $91$ prime?

(spoiler)

To determine whether $91$ is prime, we test divisibility by small prime numbers.

$91$ is not even, so it is not divisible by $2$ .
The digit sum is $9 + 1 = 10$ , so it is not divisible by $3$ .
It does not end in $0$ or $5$ , so it is not divisible by $5$ .
Try $7$ : $91 \div 7 = 13$
Since $91$ can be written as a product of two integers greater than $1$ , it is not prime.

Answer: Not prime

Sequences

Definitions

Sequence: An ordered list of numbers where each term has a specific position
Term: A number in a sequence
Index: The position number of a term in a sequence
Notation: $a_{n}$ means the $n$ -th term of the sequence $a_{1}$ is the first term, $a_{2}$ is the second term, and $a_{n}$ is the $n$ -th term

A common goal is to find a specific term, such as $a_{10}$ , or to write a rule that generates every term in the sequence.

Arithmetic sequences

An arithmetic sequence changes by adding the same constant amount each time. This constant is called the common difference and is usually written as $d$ .

Definitions

Arithmetic sequence: A sequence where the difference between consecutive terms is constant
Common difference: The constant amount added each step, written as $d$

The explicit formula for an arithmetic sequence is:

$a_{n} = a_{1} + (n - 1) d$

Example: Arithmetic sequence Sequence: $4, 9, 14, 19, \dots$

$a_{1} = 4$

$d = 5$

Example: Find the $10$ -th term

(spoiler)

$a_{10} = 4 + (10 - 1) \cdot 5 = 49$

Answer: $49$

Geometric sequences

A geometric sequence changes by multiplying by the same constant each time. This constant is called the common ratio and is usually written as $r$ .

Definitions

Geometric sequence: A sequence where the ratio between consecutive terms is constant
Common ratio: The constant multiplier each step, written as $r$

The explicit formula for a geometric sequence is:

$a_{n} = a_{1} \cdot r^{n - 1}$

Example: Sequence $5, 10, 20, 40, \dots$

$a_{1} = 5$

$r = 2$

Recursive sequences

Definitions

Recursive sequence: A sequence defined by one or more starting terms and a rule that uses earlier terms to produce later terms

A recursive definition includes:

An initial term (such as $a_{1}$ )
A recursive rule that uses previous terms to produce the next one

Recursive formulas emphasize process. To find $a_{5}$ , you compute $a_{2}$ , then $a_{3}$ , then $a_{4}$ , and then $a_{5}$ .

Example: Recursive arithmetic sequence

$a_{1} = 3$

$a_{n + 1} = a_{n} + 4$

Working out the first several terms:

$a_{1} = 3$

$a_{2} = a_{1} + 4 = 3 + 4 = 7$

$a_{3} = a_{2} + 4 = 7 + 4 = 11$

$a_{4} = a_{3} + 4 = 11 + 4 = 15$

$a_{5} = a_{4} + 4 = 15 + 4 = 19$

Answer: The sequence becomes: $3, 7, 11, 15, 19, \dots$

Example: Recursive geometric sequence

$a_{1} = 5$

$a_{n + 1} = 2 a_{n}$

Working out the first several terms:

$a_{1} = 5$

$a_{2} = 2 a_{1} = 2 \cdot 5 = 10$

$a_{3} = 2 a_{2} = 2 \cdot 10 = 20$

$a_{4} = 2 a_{3} = 2 \cdot 20 = 40$

$a_{5} = 2 a_{4} = 2 \cdot 40 = 80$

Answer: The sequence becomes: $5, 10, 20, 40, 80, \dots$

Achievable Praxis Core: Math (5733)

1. Number and quantity

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Properties of whole numbers

13 min read

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Definitions

Whole numbers: Whole numbers include $0$ and all positive integers: $0, 1, 2, 3, 4, 5, \dots$ They do not include negative numbers, fractions, or decimals.

Factors

Definitions

Factor: A factor of a whole number is a number that divides it evenly, with no remainder.

Listing factors

Before using tools like prime factorization, it helps to practice listing factors for smaller numbers.

Example: Listing factors The number $12$ has the following factors: $1, 2, 3, 4, 6, 12$ because each divides $12$ exactly.

$12 \div 1 = 12$

$12 \div 2 = 6$

$12 \div 3 = 4$

$12 \div 4 = 3$

$12 \div 6 = 2$

$12 \div 12 = 1$ Factors always come in pairs, one below the square root and one above.

Answer: $1, 2, 3, 4, 6, 12$

Factor pairs help you work efficiently. Once you’ve checked all possible factors up to the square root, you’ve automatically found the larger factors as the matching partners.

Divisibility rules

Divisibility rules let you check whether one number divides another without doing long division. They’re based on digit patterns, so they’re fast and help reduce arithmetic mistakes.

Useful rules

Divisible by 2: Last digit is $0, 2, 4, 6, 8$
Divisible by 3: Sum of digits divisible by $3$
Divisible by 4: Last two digits form a number divisible by $4$
Divisible by 5: Last digit is $0$ or $5$
Divisible by 6: Divisible by both $2$ and $3$
Divisible by 9: Sum of digits divisible by $9$
Divisible by 10: Last digit is $0$

Example: Using a divisibility rule Is $132$ divisible by $3$ ?

$1 + 3 + 2 = 6$

$6$ is divisible by $3$

Answer: Yes

These rules are especially useful when you’re simplifying fractions or breaking numbers into prime factors.

Using divisibility to find factors

Divisibility rules give you a systematic way to factor numbers: test small primes, divide, and keep going until what’s left is prime.

Example: Factoring through divisibility

Example: Find factors of $36$ We start with small prime numbers. Because $36$ is even, it is divisible by $2$ .

$36 \div 2 = 18$ This shows that $2$ and $18$ are a factor pair of $36$ .

$18 \div 2 = 9$

Dividing again by $2$ shows that $4 = 2 \times 2$ is a factor of $36$ , with $9$ remaining.

At this point, $9$ is no longer divisible by $2$ , so we switch to the next smallest prime number, $3$ .

$9 \div 3 = 3$

$3 \div 3 = 1$

This confirms that $9 = 3 \times 3$ . Combining all the prime factors used:

Prime factorization: $36 = 2^{2} \times 3^{2}$

The full list of factors comes from pairing these primes in every possible way:

Answer: Factors: $1, 2, 3, 4, 6, 9, 12, 18, 36$

Example: Find factors of $72$ We again start with the smallest prime number. Since $72$ is even, we repeatedly divide by $2$ .

$72 \div 2 = 36$

$36 \div 2 = 18$

$18 \div 2 = 9$

These steps show that $72$ contains three factors of $2$ , since $72 = 2 \times 2 \times 2 \times 9$ .

Because $9$ is no longer divisible by $2$ , we switch to $3$ .

$9 \div 3 = 3$

$3 \div 3 = 1$

This shows that $9 = 3 \times 3$ . Putting everything together:

Prime factorization: $72 = 2^{3} \times 3^{2}$

To find all factors, we form products using different combinations of the prime factors. For example:

$2 \times 2 = 4$

$2 \times 2 \times 2 = 8$

$3 \times 4 = 12$

$3 \times 6 = 18$

$3^{2} \times 2^{3} = 72$

Answer: Factors: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72$

Factor trees

Factor trees give you a visual way to break a number into primes. Each branch splits a number into smaller factors until only prime numbers remain.

Definitions

Factor tree: A diagram that splits a number into factors until only prime numbers remain.

Example: Factor tree for $36$ Prime factorization: $36 = 2^{2} \times 3^{2}$

Example: Factor tree for $60$

(spoiler)

Prime factorization: $60 = 2^{2} \times 3 \times 5$

Full factor list: $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$

Factor tree for 60

Common factors, GCF, and GCD

Definitions

Common factor: A factor shared by two or more numbers
Greatest common factor (GCF) / Greatest common divisor (GCD): The largest whole number that divides each of the given numbers evenly

A reliable method for finding the GCF or GCD:

Write prime factorizations of each number.
Identify the primes they have in common.
Use the smallest exponent of each shared prime.
Multiply these together.

Example: GCF of $48$ and $180$

(spoiler)

$48 = 2^{4} \times 3$

$180 = 2^{2} \times 3^{2} \times 5$

Shared lowest powers: $2^{2} \times 3 = 12$

Answer: $12$

Example: GCD of $252$ and $336$

(spoiler)

$252 = 2^{2} \times 3^{2} \times 7$

$336 = 2^{4} \times 3 \times 7$

GCD = $2^{2} \times 3 \times 7 = 84$

Answer: $84$

Multiples and the least common multiple (LCM)

Multiples “expand outward” from a number by repeated multiplication. The LCM is the smallest number that both values divide into evenly.

Definitions

Multiple: A number of the form $n \times k$
Common multiple: A number that is a multiple of two or more values
Least common multiple (LCM): The smallest positive common multiple

Example: LCM of $8$ and $18$

$8 = 2^{3}$

$18 = 2 \times 3^{2}$

LCM uses highest powers:

$2^{3} \times 3^{2} = 72$

Answer: $72$

Even and odd numbers

Even and odd numbers describe parity, which shows up in patterns, divisibility, and algebra.

Even: divisible by $2$ ; form $2 k$
Odd: not divisible by $2$ ; form $2 k + 1$

These definitions explain many common patterns. For example, adding two odd numbers always gives an even result, and multiplying an even number by any whole number always gives an even product.

Prime numbers

Prime numbers are the basic “atoms” of number theory. Every whole number greater than $1$ is either prime or composite.

Definitions

Prime number: A number greater than $1$ with exactly two factors: $1$ and itself

Important notes:

$2$ is the only even prime
$1$ is not prime
Composite numbers have additional factors

Testing for primality

To test whether a number is prime, check divisibility using primes up to $n$ .

Sidenote

How factor pairs work with square roots

Example: Is $37$ prime?

$37 \approx 6.08$

Test primes $2, 3, 5$

Answer: $37$ is prime

Example: Is $45$ prime?

(spoiler)

$45 \approx 6.7$

Divisible by $3$ and $5$

Answer: Not prime

More practice with divisibility and factorization

Example: Test $234$ for divisibility

(spoiler)

We check divisibility using standard rules for each number.

$2$ : The number ends in $4$ , which is even, so $234$ is divisible by $2$ .
$3$ : Add the digits: $2 + 3 + 4 = 9$ . Since $9$ is divisible by $3$ , $234$ is divisible by $3$ .
$4$ : Look at the last two digits: $34$ . Because $34$ is not divisible by $4$ , $234$ is not divisible by $4$ .
$5$ : The number does not end in $0$ or $5$ , so it is not divisible by $5$ .
$6$ : A number is divisible by $6$ if it is divisible by both $2$ and $3$ . Since both conditions are met, $234$ is divisible by $6$ .
$9$ : The digit sum is $9$ , which is divisible by $9$ , so $234$ is divisible by $9$ .
$10$ : The number does not end in $0$ , so it is not divisible by $10$ .

Answer: Divisible by $2, 3, 6,$ and $9$

Example: Prime factorization of $96$

(spoiler)

Because $96$ is even, we repeatedly divide by $2$ until no longer possible.

$96 \div 2 = 48$
$48 \div 2 = 24$
$24 \div 2 = 12$
$12 \div 2 = 6$
$6 \div 2 = 3$
At this point, $3$ is prime and cannot be divided further.
This shows that $96$ contains five factors of $2$ and one factor of $3$ .

Answer: $96 = 2^{5} \times 3$

Example: Is $91$ prime?

(spoiler)

To determine whether $91$ is prime, we test divisibility by small prime numbers.

$91$ is not even, so it is not divisible by $2$ .
The digit sum is $9 + 1 = 10$ , so it is not divisible by $3$ .
It does not end in $0$ or $5$ , so it is not divisible by $5$ .
Try $7$ : $91 \div 7 = 13$
Since $91$ can be written as a product of two integers greater than $1$ , it is not prime.

Answer: Not prime

Sequences

Definitions

Sequence: An ordered list of numbers where each term has a specific position
Term: A number in a sequence
Index: The position number of a term in a sequence
Notation: $a_{n}$ means the $n$ -th term of the sequence $a_{1}$ is the first term, $a_{2}$ is the second term, and $a_{n}$ is the $n$ -th term

A common goal is to find a specific term, such as $a_{10}$ , or to write a rule that generates every term in the sequence.

Arithmetic sequences

An arithmetic sequence changes by adding the same constant amount each time. This constant is called the common difference and is usually written as $d$ .

Definitions

Arithmetic sequence: A sequence where the difference between consecutive terms is constant
Common difference: The constant amount added each step, written as $d$

The explicit formula for an arithmetic sequence is:

$a_{n} = a_{1} + (n - 1) d$

Example: Arithmetic sequence Sequence: $4, 9, 14, 19, \dots$

$a_{1} = 4$

$d = 5$

Example: Find the $10$ -th term

(spoiler)

$a_{10} = 4 + (10 - 1) \cdot 5 = 49$

Answer: $49$

Geometric sequences

A geometric sequence changes by multiplying by the same constant each time. This constant is called the common ratio and is usually written as $r$ .

Definitions

Geometric sequence: A sequence where the ratio between consecutive terms is constant
Common ratio: The constant multiplier each step, written as $r$

The explicit formula for a geometric sequence is:

$a_{n} = a_{1} \cdot r^{n - 1}$

Example: Sequence $5, 10, 20, 40, \dots$

$a_{1} = 5$

$r = 2$

Recursive sequences

Definitions

Recursive sequence: A sequence defined by one or more starting terms and a rule that uses earlier terms to produce later terms

A recursive definition includes:

An initial term (such as $a_{1}$ )
A recursive rule that uses previous terms to produce the next one

Recursive formulas emphasize process. To find $a_{5}$ , you compute $a_{2}$ , then $a_{3}$ , then $a_{4}$ , and then $a_{5}$ .

Example: Recursive arithmetic sequence

$a_{1} = 3$

$a_{n + 1} = a_{n} + 4$

Working out the first several terms:

$a_{1} = 3$

$a_{2} = a_{1} + 4 = 3 + 4 = 7$

$a_{3} = a_{2} + 4 = 7 + 4 = 11$

$a_{4} = a_{3} + 4 = 11 + 4 = 15$

$a_{5} = a_{4} + 4 = 15 + 4 = 19$

Answer: The sequence becomes: $3, 7, 11, 15, 19, \dots$

Example: Recursive geometric sequence

$a_{1} = 5$

$a_{n + 1} = 2 a_{n}$

Working out the first several terms:

$a_{1} = 5$

$a_{2} = 2 a_{1} = 2 \cdot 5 = 10$

$a_{3} = 2 a_{2} = 2 \cdot 10 = 20$

$a_{4} = 2 a_{3} = 2 \cdot 20 = 40$

$a_{5} = 2 a_{4} = 2 \cdot 40 = 80$

Answer: The sequence becomes: $5, 10, 20, 40, 80, \dots$

Factors divide numbers evenly, and prime factorization reveals the building blocks of whole numbers.
The greatest common factor (GCF) or greatest common divisor (GCD) is the largest shared divisor and is essential for simplifying fractions and ratios.
The least common multiple (LCM) identifies the smallest shared multiple and is important for matching denominators.
Parity (even or odd) and prime structure help predict how numbers behave.
Sequences describe numerical patterns: arithmetic uses repeated addition, geometric uses repeated multiplication, and recursive formulas show how each term follows from the previous one.

Properties of whole numbers

Definitions

Whole numbers: Whole numbers include $0$ and all positive integers: $0, 1, 2, 3, 4, 5, \dots$ They do not include negative numbers, fractions, or decimals.

Factors

Definitions

Factor: A factor of a whole number is a number that divides it evenly, with no remainder.

Listing factors

Before using tools like prime factorization, it helps to practice listing factors for smaller numbers.

Example: Listing factors The number $12$ has the following factors: $1, 2, 3, 4, 6, 12$ because each divides $12$ exactly.

$12 \div 1 = 12$

$12 \div 2 = 6$

$12 \div 3 = 4$

$12 \div 4 = 3$

$12 \div 6 = 2$

$12 \div 12 = 1$ Factors always come in pairs, one below the square root and one above.

Answer: $1, 2, 3, 4, 6, 12$

Factor pairs help you work efficiently. Once you’ve checked all possible factors up to the square root, you’ve automatically found the larger factors as the matching partners.

Divisibility rules

Divisibility rules let you check whether one number divides another without doing long division. They’re based on digit patterns, so they’re fast and help reduce arithmetic mistakes.

Useful rules

Divisible by 2: Last digit is $0, 2, 4, 6, 8$
Divisible by 3: Sum of digits divisible by $3$
Divisible by 4: Last two digits form a number divisible by $4$
Divisible by 5: Last digit is $0$ or $5$
Divisible by 6: Divisible by both $2$ and $3$
Divisible by 9: Sum of digits divisible by $9$
Divisible by 10: Last digit is $0$

Example: Using a divisibility rule Is $132$ divisible by $3$ ?

$1 + 3 + 2 = 6$

$6$ is divisible by $3$

Answer: Yes

These rules are especially useful when you’re simplifying fractions or breaking numbers into prime factors.

Using divisibility to find factors

Divisibility rules give you a systematic way to factor numbers: test small primes, divide, and keep going until what’s left is prime.

Example: Factoring through divisibility

Example: Find factors of $36$ We start with small prime numbers. Because $36$ is even, it is divisible by $2$ .

$36 \div 2 = 18$ This shows that $2$ and $18$ are a factor pair of $36$ .

$18 \div 2 = 9$

Dividing again by $2$ shows that $4 = 2 \times 2$ is a factor of $36$ , with $9$ remaining.

At this point, $9$ is no longer divisible by $2$ , so we switch to the next smallest prime number, $3$ .

$9 \div 3 = 3$

$3 \div 3 = 1$

This confirms that $9 = 3 \times 3$ . Combining all the prime factors used:

Prime factorization: $36 = 2^{2} \times 3^{2}$

The full list of factors comes from pairing these primes in every possible way:

Answer: Factors: $1, 2, 3, 4, 6, 9, 12, 18, 36$

Example: Find factors of $72$ We again start with the smallest prime number. Since $72$ is even, we repeatedly divide by $2$ .

$72 \div 2 = 36$

$36 \div 2 = 18$

$18 \div 2 = 9$

These steps show that $72$ contains three factors of $2$ , since $72 = 2 \times 2 \times 2 \times 9$ .

Because $9$ is no longer divisible by $2$ , we switch to $3$ .

$9 \div 3 = 3$

$3 \div 3 = 1$

This shows that $9 = 3 \times 3$ . Putting everything together:

Prime factorization: $72 = 2^{3} \times 3^{2}$

To find all factors, we form products using different combinations of the prime factors. For example:

$2 \times 2 = 4$

$2 \times 2 \times 2 = 8$

$3 \times 4 = 12$

$3 \times 6 = 18$

$3^{2} \times 2^{3} = 72$

Answer: Factors: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72$

Factor trees

Factor trees give you a visual way to break a number into primes. Each branch splits a number into smaller factors until only prime numbers remain.

Definitions

Factor tree: A diagram that splits a number into factors until only prime numbers remain.

Example: Factor tree for $36$ Prime factorization: $36 = 2^{2} \times 3^{2}$

Example: Factor tree for $60$

(spoiler)

Prime factorization: $60 = 2^{2} \times 3 \times 5$

Full factor list: $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$

Common factors, GCF, and GCD

Definitions

Common factor: A factor shared by two or more numbers
Greatest common factor (GCF) / Greatest common divisor (GCD): The largest whole number that divides each of the given numbers evenly

A reliable method for finding the GCF or GCD:

Write prime factorizations of each number.
Identify the primes they have in common.
Use the smallest exponent of each shared prime.
Multiply these together.

Example: GCF of $48$ and $180$

(spoiler)

$48 = 2^{4} \times 3$

$180 = 2^{2} \times 3^{2} \times 5$

Shared lowest powers: $2^{2} \times 3 = 12$

Answer: $12$

Example: GCD of $252$ and $336$

(spoiler)

$252 = 2^{2} \times 3^{2} \times 7$

$336 = 2^{4} \times 3 \times 7$

GCD = $2^{2} \times 3 \times 7 = 84$

Answer: $84$

Multiples and the least common multiple (LCM)

Multiples “expand outward” from a number by repeated multiplication. The LCM is the smallest number that both values divide into evenly.

Definitions

Multiple: A number of the form $n \times k$
Common multiple: A number that is a multiple of two or more values
Least common multiple (LCM): The smallest positive common multiple

Example: LCM of $8$ and $18$

$8 = 2^{3}$

$18 = 2 \times 3^{2}$

LCM uses highest powers:

$2^{3} \times 3^{2} = 72$

Answer: $72$

Even and odd numbers

Even and odd numbers describe parity, which shows up in patterns, divisibility, and algebra.

Even: divisible by $2$ ; form $2 k$
Odd: not divisible by $2$ ; form $2 k + 1$

These definitions explain many common patterns. For example, adding two odd numbers always gives an even result, and multiplying an even number by any whole number always gives an even product.

Prime numbers

Prime numbers are the basic “atoms” of number theory. Every whole number greater than $1$ is either prime or composite.

Definitions

Prime number: A number greater than $1$ with exactly two factors: $1$ and itself

Important notes:

$2$ is the only even prime
$1$ is not prime
Composite numbers have additional factors

Testing for primality

To test whether a number is prime, check divisibility using primes up to $n$ .

Sidenote

How factor pairs work with square roots

Example: Is $37$ prime?

$37 \approx 6.08$

Test primes $2, 3, 5$

Answer: $37$ is prime

Example: Is $45$ prime?

(spoiler)

$45 \approx 6.7$

Divisible by $3$ and $5$

Answer: Not prime

More practice with divisibility and factorization

Example: Test $234$ for divisibility

(spoiler)

We check divisibility using standard rules for each number.

$2$ : The number ends in $4$ , which is even, so $234$ is divisible by $2$ .
$3$ : Add the digits: $2 + 3 + 4 = 9$ . Since $9$ is divisible by $3$ , $234$ is divisible by $3$ .
$4$ : Look at the last two digits: $34$ . Because $34$ is not divisible by $4$ , $234$ is not divisible by $4$ .
$5$ : The number does not end in $0$ or $5$ , so it is not divisible by $5$ .
$6$ : A number is divisible by $6$ if it is divisible by both $2$ and $3$ . Since both conditions are met, $234$ is divisible by $6$ .
$9$ : The digit sum is $9$ , which is divisible by $9$ , so $234$ is divisible by $9$ .
$10$ : The number does not end in $0$ , so it is not divisible by $10$ .

Answer: Divisible by $2, 3, 6,$ and $9$

Example: Prime factorization of $96$

(spoiler)

Because $96$ is even, we repeatedly divide by $2$ until no longer possible.

$96 \div 2 = 48$
$48 \div 2 = 24$
$24 \div 2 = 12$
$12 \div 2 = 6$
$6 \div 2 = 3$
At this point, $3$ is prime and cannot be divided further.
This shows that $96$ contains five factors of $2$ and one factor of $3$ .

Answer: $96 = 2^{5} \times 3$

Example: Is $91$ prime?

(spoiler)

To determine whether $91$ is prime, we test divisibility by small prime numbers.

$91$ is not even, so it is not divisible by $2$ .
The digit sum is $9 + 1 = 10$ , so it is not divisible by $3$ .
It does not end in $0$ or $5$ , so it is not divisible by $5$ .
Try $7$ : $91 \div 7 = 13$
Since $91$ can be written as a product of two integers greater than $1$ , it is not prime.

Answer: Not prime

Sequences

Definitions

Sequence: An ordered list of numbers where each term has a specific position
Term: A number in a sequence
Index: The position number of a term in a sequence
Notation: $a_{n}$ means the $n$ -th term of the sequence $a_{1}$ is the first term, $a_{2}$ is the second term, and $a_{n}$ is the $n$ -th term

A common goal is to find a specific term, such as $a_{10}$ , or to write a rule that generates every term in the sequence.

Arithmetic sequences

An arithmetic sequence changes by adding the same constant amount each time. This constant is called the common difference and is usually written as $d$ .

Definitions

Arithmetic sequence: A sequence where the difference between consecutive terms is constant
Common difference: The constant amount added each step, written as $d$

The explicit formula for an arithmetic sequence is:

$a_{n} = a_{1} + (n - 1) d$

Example: Arithmetic sequence Sequence: $4, 9, 14, 19, \dots$

$a_{1} = 4$

$d = 5$

Example: Find the $10$ -th term

(spoiler)

$a_{10} = 4 + (10 - 1) \cdot 5 = 49$

Answer: $49$

Geometric sequences

A geometric sequence changes by multiplying by the same constant each time. This constant is called the common ratio and is usually written as $r$ .

Definitions

Geometric sequence: A sequence where the ratio between consecutive terms is constant
Common ratio: The constant multiplier each step, written as $r$

The explicit formula for a geometric sequence is:

$a_{n} = a_{1} \cdot r^{n - 1}$

Example: Sequence $5, 10, 20, 40, \dots$

$a_{1} = 5$

$r = 2$

Recursive sequences

Definitions

Recursive sequence: A sequence defined by one or more starting terms and a rule that uses earlier terms to produce later terms

A recursive definition includes:

An initial term (such as $a_{1}$ )
A recursive rule that uses previous terms to produce the next one

Recursive formulas emphasize process. To find $a_{5}$ , you compute $a_{2}$ , then $a_{3}$ , then $a_{4}$ , and then $a_{5}$ .

Example: Recursive arithmetic sequence

$a_{1} = 3$

$a_{n + 1} = a_{n} + 4$

Working out the first several terms:

$a_{1} = 3$

$a_{2} = a_{1} + 4 = 3 + 4 = 7$

$a_{3} = a_{2} + 4 = 7 + 4 = 11$

$a_{4} = a_{3} + 4 = 11 + 4 = 15$

$a_{5} = a_{4} + 4 = 15 + 4 = 19$

Answer: The sequence becomes: $3, 7, 11, 15, 19, \dots$

Example: Recursive geometric sequence

$a_{1} = 5$

$a_{n + 1} = 2 a_{n}$

Working out the first several terms:

$a_{1} = 5$

$a_{2} = 2 a_{1} = 2 \cdot 5 = 10$

$a_{3} = 2 a_{2} = 2 \cdot 10 = 20$

$a_{4} = 2 a_{3} = 2 \cdot 20 = 40$

$a_{5} = 2 a_{4} = 2 \cdot 40 = 80$

Answer: The sequence becomes: $5, 10, 20, 40, 80, \dots$