Properties of whole numbers

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. The chapter closes with sequences, number patterns built by a repeated rule.

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

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.

Factors come in pairs whose product equals the original number. 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 8: Last three digits form a number divisible by $8$
• Divisible by 9: Sum of digits divisible by $9$
• Divisible by 10: Last digit is $0$

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

Remainders

When a number doesn’t divide evenly, the amount left over is the remainder: $23÷5=4$ remainder $3$, because $23=5\times 4+3$. So a number that leaves remainder $r$ when divided by $d$ is $r$ more than a multiple of $d$.

Example: two conditions at once

A number of marbles splits evenly into bags of $6$, but bagging them by $5$ leaves $2$ over. Could there be $42$, $48$, or $54$ marbles?

• All three are multiples of $6$, so check the remainder when each is divided by $5$.
• $42=5\times 8+2$ leaves $2$; $48$ leaves $3$; $54$ leaves $4$.

Answer: $42$

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: find factors of $36$

We start with small prime numbers. Because $36$ is even, it is divisible by $2$.

• $36÷2=18$ - this shows that $2$ and $18$ are a factor pair of $36$.
• $18÷2=9$ - dividing again by $2$ shows that $36=2\times 2\times 9$.

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

• $9÷3=3$
• $3÷3=1$

This confirms that $9=3\times 3$. Combining all the prime factors used gives the prime factorization: $36=2^2\times 3^2$

To list all factors, find every pair that multiplies to $36$: $(1,36)$, $(2,18)$, $(3,12)$, $(4,9)$, $(6,6)$.

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

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 $60$

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

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:

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

Watch out - GCF vs. LCM: Both use prime factorizations, but they work in opposite directions. For the GCF, take only the shared primes and use the smallest exponent of each. For the LCM, take all primes that appear (in either number) and use the largest exponent of each. Mixing these up is one of the most common mistakes on this topic.

Example: GCF of $48$ and $180$

• $48=2^4\times 3$
• $180=2^2\times 3^2\times 5$
• Shared lowest powers: $2^2\times 3=12$

Answer: $12$

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 multiple of $n$ is a product $n\times k$ where $k$ is a positive integer. The multiples of $5$ are $5,10,15,20,\dots$

Common multiple

A number that is a multiple of two or more values

Least common multiple (LCM)

The smallest positive common multiple

Factors and multiples work in opposite directions: the factors of $12$ are $1,2,3,4,6,12$ - they all divide in evenly and are $\le 12$. The multiples of $12$ are $12,24,36,48,\dots$ - they are built by multiplying out and are all $\ge 12$. A factor divides in; a multiple is built out.

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

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

Common pitfalls with prime numbers:

• $1$ is not prime. It has only one factor (itself), so it doesn’t meet the definition.
• $2$ is the only even prime. Every other even number is divisible by $2$, giving it at least three factors.

Testing for primality

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

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 $\sqrt{N}$. Once you pass $\sqrt{N}$, all larger factors already appeared as the “other half” of a smaller factor.

So checking primes up to $\sqrt{n}$ is enough to confirm whether a number is prime.

Example: is $37$ prime?

• $\sqrt{37}\approx 6.08$, so we only need to test primes up to $6$: that’s $2$, $3$, and $5$.
• $37$ is odd, so it’s not divisible by $2$.
• Digit sum: $3+7=10$, which is not divisible by $3$.
• $37$ doesn’t end in $0$ or $5$, so it’s not divisible by $5$.
• No prime up to $\sqrt{37}$ divides $37$.

Answer: $37$ is prime

Sequences

A sequence is a list of numbers in a set order, and each number is a term. Write $a_1$ for the first term and $a_n$ for the term in position $n$. A recursive rule says how to get the next term, $a_{n+1}$, from the one before it.

Definitions

Arithmetic sequence

Each term is the one before plus the same amount, the common difference $d$: $a_{n+1}=a_n+d$.

Geometric sequence

Each term is the one before times the same amount, the common ratio $r$: $a_{n+1}=r\cdot a_n$.

To tell which you have, check the differences, then the ratios. In $3,7,11,15,\dots$ every difference is $4$, so $a_{n+1}=a_n+4$. In $2,6,18,54,\dots$ the differences change, but every ratio is $3$, so $a_{n+1}=3a_n$.

To reach a far-off term without listing every one, count the steps: the $n$th term is $n-1$ steps past the first.

$$a_n=a_1+(n-1)d{a}_n=a_1\cdot r^{n-1}$$

Example: finding a later term

(a) An arithmetic sequence begins $7,12,17$. What is the $20$th term?

(b) A sequence begins $3,-6,12$. What is the $6$th term?

• (a) $d=5$, and the $20$th term is $19$ steps past the first: $7+19\cdot 5=102$.
• (b) Each term is the one before times $-2$, so the signs alternate: $3\cdot (-2{)}^5=-96$.

Answer: (a) $102$; (b) $-96$

Some rules use more than one earlier term, such as “each term is the sum of the two before it.” Apply the rule one term at a time and stop at the term the question asks for. A rule also runs backward: if each term is the sum of the three before it, then in $a,b,c,10,17,33$ the last term gives $33=c+10+17$, so $c=6$.

Example: a rule that uses two earlier terms

Each term after the first two is the sum of the two before it: $2,5,7,x,y,z$. Find $z$.

• $x=5+7=12$
• $y=7+12=19$
• $z=12+19=31$

Answer: $31$