Factors and multiples

This chapter applies to all AMC 8/10/12 test takers.

Divisors (also called factors) and multiples show up constantly on the AMC.

• A divisor of a number is a whole number that divides it evenly (with no remainder).
  • Example: $3$ is a divisor of $12$ because $12÷3=4$.
  • Non-example: $3$ is not a divisor of $13$ because $13÷3$ leaves a remainder.
• A multiple of a number is what you get when you multiply it by a whole number.
  • Example: $24$ is a multiple of $12$ because $2\times 12=24$.

Two quick facts to keep in mind:

• Every positive integer has $1$ as a factor.
• Every positive integer has itself as a multiple (because $1$ times the number is the number).

Factors (divisors)

Divisibility rules let you test quickly whether a number has certain factors. Knowing these rules helps you find factors efficiently on test day.

Factors	Divisibility rules
$2$	All even numbers are divisible by $2$.
$3$	A number is divisible by $3$ if the sum of its digits is a multiple of $3$.
$4$	A number is divisible by $4$ if its last two digits form a number divisible by $4$.
$5$	All numbers that end with a $0$ or $5$ are divisible by $5$.
$6$	A number is divisible by $6$ if it is divisible by $2$ and by $3$ (equivalently: even, and its digit sum is a multiple of $3$).
$8$	A number is divisible by $8$ if its last three digits form a number divisible by $8$.
$9$	A number is divisible by $9$ if the sum of its digits is a multiple of $9$.

Example: The question below is from 2022 AMC 10B

How many of the first ten numbers of the sequence $121,11211,1112111,\dots$ are prime numbers?
A. $0$
B. $1$
C. $2$
D. $3$
E. $4$

If you’d like to review how to find the prime factors of a number, see the prime numbers chapter.

To find the total number of unique factors quickly:

• Find the prime factorization.
• Add $1$ to each exponent.
• Multiply the results.

For example, the prime factorization of $14$ is $2^1(7^1)$. The exponents are $1$ and $1$. Adding $1$ to each exponent gives $2$ and $2$, and multiplying gives $4$. So $14$ has $4$ unique factors: $1$, $2$, $7$, and $14$.

Multiples

Multiples are found by multiplying the number by any positive integer. This is essentially the opposite of factors.

Here are some useful multiple rules to keep in mind.

Multiple rules
All multiples of even numbers must also be even.
The multiple of any number that ends in $0$ must also end in $0$.
The unit digit of concurrent multiples always follow a specific pattern solely based on the unit digit of the original number.
All multiples of numbers that end in $5$ must end in $0$ or $5$.

Greatest common factors and least common multiples

The least common multiple (LCM) of two (or more) numbers is the smallest positive multiple they share.

• Example: The LCM of $2$ and $3$ is $6$.
  • Multiples of $2$: $2,4,6,8,\dots$
  • Multiples of $3$: $3,6,9,12,\dots$
  • The first shared multiple is $6$.

Every common multiple of $2$ and $3$ is also a multiple of $6$.

The greatest common factor (GCF), also known as the greatest common denominator (GCM), is the largest factor shared by two (or more) numbers.

• Example: The GCF of $12$ and $8$ is $4$.
  • Factors of $12$: $1,2,3,4,6,12$
  • Factors of $8$: $1,2,4,8$
  • The largest shared factor is $4$.

A reliable way to find both the GCF and LCM (even for large numbers) is to use prime factorizations. We’ll use the same two numbers to find both.

First, write each number as a product of primes.

$$60=2^2(3^1)(5^1)$$

$$48=2^4(3^1)$$

Finding the GCF

To find the GCF, take the smallest power of each prime that appears in both factorizations.

• Shared primes: $2$ and $3$
• Smallest powers: $2^2$ and $3^1$

Multiply them.

GCD $=2^2(3^1)=12$

Finding the LCM

To find the LCM, take the greatest power of each prime that appears in either factorization.

• Primes involved: $2,3,5$
• Greatest powers: $2^4$, $3^1$, and $5^1$

Multiply them.

LCM $=2^4(3^1)(5^1)=240$

Example: The question below is from 2018 AMC 10A

Let$a,b,c,$ and $d$ be positive integers such that $\gcd (a,b)=24$, $\gcd (b,c)=36$, $\gcd (c,d)=54$, and $70<\gcd (d,a)<100$. Which of the following must be a divisor of $a$? (gcd means greatest common factor)
A. $5$
B. $7$
C. $11$
D. $13$
E. $17$

Common themes

• The product of any two integers is also the product of their LCM and GCF. $LCM(a,b)\times GCF(a,b)=a\times b$
• Only perfect squares have an odd number of factors. All other numbers, including primes, have an even number of factors.
• $a^x-b^x$ must be divisible by $a-b$.
• Always look at the unit’s digit of answer choices to determine if any of them can be automatically eliminated based on patterns of the unit digit of concurrent multiples of a number.