Find Multiples and Factors

Identify multiples of numbers

If you count shoes in pairs — $2, 4, 6, 8, 10, 12$ — each number in that list is a multiple of $2$: the product of a counting number and $2$ ($1 \cdot 2, 2 \cdot 2, 3 \cdot 2, \dots$).

Some multiples are easy to spot from a number’s last digit:

• A number is a multiple of $2$ if its last digit is $0, 2, 4, 6,$ or $8$.
• A number is a multiple of $5$ if its last digit is $5$ or $0$.
• A number is a multiple of $10$ if its last digit is $0$.

Example. Is $3{,}714$ a multiple of $2$? Its last digit is $4$, one of $0, 2, 4, 6, 8$ — yes.

Example. Is $880$ a multiple of $5$? Its last digit is $0$ — yes.

The pattern for multiples of $3$ is different — it isn’t about the last digit, but the sum of the digits: if the sum of a number’s digits is itself a multiple of $3$, the number is a multiple of $3$.

Example. Is $645$ a multiple of $3$? Sum the digits: $6 + 4 + 5 = 15$, and $15$ is a multiple of $3$ — yes. Check: $645 \div 3 = 215$.

Example. Is $10{,}519$ a multiple of $3$? Sum the digits: $1 + 0 + 5 + 1 + 9 = 16$, not a multiple of $3$ — no. Dividing confirms it: $10{,}519 \div 3 = 3{,}506\ \text{R}1$, not a whole number.

Use common divisibility tests

If a number $m$ is a multiple of $n$, we also say $m$ is divisible by $n$ — multiplication and division are inverse operations, so the multiple patterns above double as divisibility tests.

Example. Is $1{,}290$ divisible by $2, 3, 5,$ and $10$?

• By $2$: last digit $0$ — yes ($1290 \div 2 = 645$).
• By $3$: $1 + 2 + 9 + 0 = 12$, divisible by $3$ — yes ($1290 \div 3 = 430$).
• By $5$: last digit $0$ — yes ($1290 \div 5 = 258$).
• By $10$: last digit $0$ — yes ($1290 \div 10 = 129$).

So $1{,}290$ is divisible by all four.

Example. Is $5{,}625$ divisible by $2, 3, 5,$ and $10$?

• By $2$: last digit $5$ — no.
• By $3$: $5 + 6 + 2 + 5 = 18$, divisible by $3$ — yes.
• By $5$: last digit $5$ — yes.
• By $10$: last digit $5$, not $0$ — no.

So $5{,}625$ is divisible by $3$ and $5$, but not $2$ or $10$.

Find all the factors of a number

If $72$ is the product of $8$ and $9$, we can also say $8$ and $9$ are factors of $72$, and that we have factored $72$: $8 \cdot 9 = 72$.

We can find every factor of a counting number by dividing it by each counting number in turn, starting at $1$, stopping once the quotient becomes smaller than the divisor (past that point the same factor pairs would just repeat in reverse order).

Example. Find all the factors of $72$.

The next divisor, $9$, gives quotient $8$ — smaller than the divisor, so we stop. All the factors of $72$, smallest to largest:

Identify prime and composite numbers

Some numbers, like $72$, have many factors. Others, like $7$, have only two: $1$ and the number itself.

The prime numbers less than $20$ are $2, 3, 5, 7, 11, 13, 17,$ and $19$. To test whether a larger number is prime, check whether any prime number divides it, testing primes in order and stopping once the quotient drops below the divisor.

Example. Is $83$ prime or composite? Testing primes in order: not divisible by $2$ (last digit isn’t even), not by $3$ ($8+3=11$, not a multiple of $3$), not by $5$ (last digit isn’t $5$ or $0$), not by $7$ ($83 \div 7 \approx 11.857$), not by $11$ ($83 \div 11 \approx 7.545$, and now the quotient is smaller than the divisor, so we stop). No prime factors were found, so $83$ is prime.

Example. Is $77$ prime or composite? Not divisible by $2$, $3$, or $5$ — but $77 \div 7 = 11$, a whole number. Since $7$ is a prime factor, $77$ is composite.

Key terms

multiple of a number — the product of a counting number and a given number. divisible — $m$ is divisible by $n$ if $m$ is a multiple of $n$. factors — in $a \cdot b = m$, both $a$ and $b$ are factors of $m$. prime number — a counting number greater than $1$ whose only factors are $1$ and itself. composite number — a counting number that is not prime.

Practice

Identify multiples of numbers

Use common divisibility tests

Find all the factors of a number

Identify prime and composite numbers

This section is adapted from Prealgebra 2e, Section 2.4: Find Multiples and Factors by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: condensed prose, recreated the multiples charts and divisibility tests as accessible Markdown tables, converted practice problems (“Try Its”) into interactive exercises with instant feedback, and adapted selected end-of-section exercises into the interactive Practice block.