Parent guide · 5 min read
Prime factorization for kids: breaking a number down to its building blocks
Prime factorization means writing a number as a product of primes only, such as 60 = 2 × 2 × 3 × 5. It sounds abstract, and children often meet it as a diagram to copy, a factor tree, without being sure what the diagram is for. This guide explains what a prime is, shows how to build a tree that cannot go wrong, and works through three examples, including a number that turns out to be prime already.
Try this tonight
4 things you can do at home
Learn the first few primes by heart
The primes below 20 are 2, 3, 5, 7, 11, 13, 17 and 19. Knowing them means a child can see at a glance when a branch is finished. Remind them that 1 is not prime, and 2 is the only even prime.
Start with the smallest prime that divides the number
Test 2, then 3, then 5, then 7. Dividing by the smallest prime that fits each time gives a tidy, ordered list and fewer missed factors than splitting into any pair that comes to mind.
Circle each prime as it appears
On a factor tree, circle every prime as soon as it shows up and stop splitting that branch. Children who keep splitting a prime run into "7 = 1 × 7" and get stuck.
Multiply back to check
Multiply all the circled primes together. If the product is the starting number, the factorization is correct. If not, a factor was dropped or a branch was not finished. It takes ten seconds and catches most errors.
What "prime" and "composite" mean
A prime number has exactly two different factors: 1 and itself. 7 is prime because only 1 × 7 makes 7. A composite number has more than two factors, such as 12 (1, 2, 3, 4, 6, 12). The number 1 is neither, because it has only one factor.
Prime factorization uses the primes as building blocks. Just as every molecule is built from atoms, every whole number above 1 is a product of primes in exactly one way, apart from the order they are written in.
A worked example: 60
Split 60 into 6 × 10. Then 6 splits into 2 × 3, and 10 splits into 2 × 5. Every branch now ends in a prime, so 60 = 2 × 2 × 3 × 5, which is written 2² × 3 × 5 using an exponent for the repeated 2. Multiplying back, 4 × 3 × 5 = 60.
The tree could equally start with 60 = 4 × 15, or 2 × 30, and the primes at the bottom would still be 2, 2, 3 and 5. That is a good point to show a child who worries they have split it "the wrong way".
A worked example: 84, using repeated division
Dividing by the smallest prime each time: 84 ÷ 2 = 42, 42 ÷ 2 = 21, 21 ÷ 3 = 7, and 7 is prime. So 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7. Writing the divisions in a column, one prime per line, gives the same answer as a tree for children who find trees messy.
A worked example: is 97 prime?
To test 97, try dividing by the primes 2, 3, 5 and 7. It is odd, its digits add to 16 so it is not a multiple of 3, it does not end in 0 or 5, and 7 × 14 = 98 is just past it, so 7 does not fit either. You only need to test primes up to the square root of 97, which is just under 10, so 97 is prime. Its factorization is just 97.
Factorizations are a tool for later work. Writing 36 = 2² × 3² and 60 = 2² × 3 × 5 makes the greatest common factor (2² × 3 = 12) and the lowest common multiple (2² × 3² × 5 = 180) of 36 and 60 easy to read off.
Signs it’s time for outside help
When home help has done what it can
- Counts 1 as a prime number, or says that 9 or 15 is prime.
- Keeps splitting a prime, writing 7 as 1 × 7 and never finishing.
- Stops while a composite number such as 6 or 10 is still on a branch.
- Gets a different answer from a different tree and thinks one of them must be wrong.
- Forgets to multiply back and cannot tell whether the factorization is correct.
- Slows down on 2, 3, 5 and 7 times-table facts, so each split takes a long time.
Questions parents ask
FAQ
What grade learns prime factorization?
Primes and composites are introduced around Grade 4 (ages 9–10) in the US Common Core, and prime factorization is used for greatest common factors and lowest common multiples in Grades 5–6. Schools vary in when exactly the term appears.
Is 1 a prime number?
No. A prime has exactly two factors, and 1 has only one. This convention is also what keeps the prime factorization of each number unique.
Does it matter how I draw the factor tree?
No. Different trees give the same set of primes at the bottom, though they may be written in a different order. Rewriting the primes from smallest to largest makes answers easy to compare.
How do I know when a number is prime?
Try dividing by the primes 2, 3, 5, 7 and so on, up to the square root of the number. If none divides it exactly, the number is prime.
What are the divisibility rules worth knowing?
A number is divisible by 2 if it ends in an even digit, by 5 if it ends in 0 or 5, by 3 if its digits add to a multiple of 3, and by 10 if it ends in 0. These make the first few splits much quicker.
Further reading
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