Lesson 4: Find Multiples and Factors

New Concept

This lesson explores how numbers are related through multiplication. You'll learn to find multiples and factors, use divisibility shortcuts, and classify numbers as prime or composite, revealing the fundamental structure of numbers.

What’s next

Next, you'll apply these ideas with interactive examples and practice cards to master identifying multiples, factors, and prime numbers.

Property

A number is a multiple of $n$ if it is the product of a counting number and $n$. A multiple of a number is the product of the number and a counting number.

Examples

• The first five multiples of 6 are found by multiplying 6 by 1, 2, 3, 4, and 5. The multiples are 6, 12, 18, 24, and 30.

• To check if 48 is a multiple of 6, we see if any counting number times 6 equals 48. Since $6 \cdot 8 = 48$, we know 48 is a multiple of 6.

Property

If a number $m$ is a multiple of $n$, then we say that $m$ is divisible by $n$. We can use divisibility tests as shortcuts.

A number is divisible by:

• 2 if the last digit is 0, 2, 4, 6, or 8
• 3 if the sum of the digits is divisible by 3
• 5 if the last digit is 5 or 0
• 6 if divisible by both 2 and 3
• 10 if the last digit is 0

Examples

• Is 738 divisible by 3? First, sum the digits: $7 + 3 + 8 = 18$. Since 18 is divisible by 3 ($18 \div 3 = 6$), the number 738 is divisible by 3.

Property

If $a \cdot b = m$, and both $a$ and $b$ are integers, then $a$ and $b$ are factors of $m$, and $m$ is the product of $a$ and $b$.

To find all the factors of a counting number:

1. Divide the number by each counting number (1, 2, 3, ...), in order, until the quotient is smaller than the divisor. If the quotient is a counting number, the divisor and quotient are a factor pair.
2. List all the factor pairs.
3. Write all the factors in order from smallest to largest.

Examples

• To find all the factors of 24, we look for pairs of numbers that multiply to 24: $1 \cdot 24$, $2 \cdot 12$, $3 \cdot 8$, and $4 \cdot 6$. So, the factors are 1, 2, 3, 4, 6, 8, 12, and 24.

Property

A prime number is a counting number greater than 1 whose only factors are 1 and itself.
A composite number is a counting number that is not prime. The number 1 is neither prime nor composite.

To determine if a number is prime:

1. Test each of the primes (2, 3, 5, ...), in order, to see if it is a factor of the number.
2. Stop when the quotient is smaller than the divisor or when a prime factor is found.
3. If it has a prime factor, it is composite. If not, it is prime.

Examples

• Is 29 a prime number? It is not divisible by 2 (it is odd), 3 (sum of digits is 11), or 5. Testing 7 gives $29 \div 7$, which has a remainder. Since the next quotient is smaller than the divisor, 29 is prime.