Factors, multiples and primes

What is this skill?

Factors and multiples are two sides of multiplication. The factors of a number are the whole numbers that divide it exactly, in pairs that multiply to make it: the factors of 18 are 1, 2, 3, 6, 9 and 18, because 1 × 18, 2 × 9 and 3 × 6 all equal 18. The multiples of a number are what you get by counting by it: the multiples of 4 are 4, 8, 12, 16 and so on.

Fourth graders also sort whole numbers by how many factors they have. A prime number has exactly two factors, 1 and itself, such as 7 or 59. A composite number has more than two, such as 8 or 35. The number 1 is neither prime nor composite, because it has just one factor. Children find factors by testing numbers in order and working in pairs, and they find multiples by skip counting.

Why it matters

Factors are the tool for simplifying fractions, finding common denominators, and later for ratios and algebra. When a child needs to rewrite 6/8 as 3/4, they are using a factor that both numbers share. When they add fractions with different denominators, they look for a common multiple.

It also deepens multiplication fluency, because every question asks the child to move easily between a product and the facts that make it. Prime numbers are a first taste of how numbers are built from smaller pieces, an idea that returns in middle school.

Common mistakes

• Mixing up factor and multiple. A helpful line: factors are few and small (they fit into the number), while multiples go on forever.
• Stopping the list too early and missing a pair, such as listing 1, 2, 3, 6 for 18 and leaving out 9 and 18.
• Calling every odd number prime. 9, 15 and 35 are odd but composite.
• Calling 2 composite because it is even. 2 is the only even prime number.
• Forgetting that every number is a factor of itself and that 1 is a factor of every number.

How to help at home

• Find factor pairs with objects: arrange 12 coins into every possible rectangle (1 × 12, 2 × 6, 3 × 4) and read off the factors.
• Skip count aloud by a chosen number, then ask whether a larger number is on the list, such as “Is 42 a multiple of 6?”
• Teach the quick divisibility checks: even numbers end in 0, 2, 4, 6 or 8; multiples of 5 end in 0 or 5; multiples of 10 end in 0.
• Play prime or composite: say a number up to 50 and let your child decide, explaining with a factor pair.
• Write the numbers 1 to 50 and shade the multiples of 3, then the multiples of 4, and talk about the overlaps.

Worked examples

These are real questions from the Tutorsaurus factors, multiples and primes practice, with the answers worked out.

1. Example 1

   Is 7 prime or composite?

   Choices: Prime, Composite

   Answer: Prime. 7 is prime: its only factors are 1 and 7.

2. Example 2

   Is 3 prime or composite?

   Choices: Prime, Composite

   Answer: Prime. 3 is prime: its only factors are 1 and 3.

3. Example 3

   Is 59 prime or composite?

   Choices: Prime, Composite

   Answer: Prime. 59 is prime: its only factors are 1 and 59.