Prime Factorization

What is a prime number again?

A prime number is a whole number greater than 1 that cannot be split into smaller whole-number factors, except by 1 and itself. The primes are 2, 3, 5, 7, 11, 13, and so on. A number like 12 is not prime, because it is 3 × 4.

What is a factor tree?

A factor tree breaks a number into factors until nothing is left but primes. Start with 60. Split it into 6 × 10. Split 6 into 2 × 3, and 10 into 2 × 5. Now every number at the bottom is prime.

Those primes are the prime factorization: 60 = 2 × 2 × 3 × 5.

Every whole number greater than 1 has exactly one prime factorization. The order of the primes can change, but the primes themselves cannot. If you start 60 as 2 × 30 instead, you still finish with 2 × 2 × 3 × 5.

If a number is already prime, the tree is just that number. 97 is prime, so its factorization is 97.

How the primes give the GCF and LCM

Once you have two numbers' primes, the greatest common factor and least common multiple fall out of them.

The greatest common factor uses the primes both numbers share, at their lowest power. Take 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. The shared primes are 2 and 3, so the GCF is 2 × 3 = 6.

The least common multiple uses every prime that appears, at its highest power. Here that is 2 × 2 and 3 × 3, so the LCM is 4 × 9 = 36.

What they're learning

• Every whole number greater than 1 is either prime or built from primes.
• A factor tree breaks a number down until only primes are left.
• The prime factorization of a number is unique.
• The shared primes give the GCF, and the highest powers give the LCM.

Aliases

• Factor trees
• Prime factors
• Prime factorization
• GCF and LCM from primes

Refresher

A prime number cannot be split into smaller whole numbers except by 1 and itself. A factor tree keeps splitting a number until only primes are left. Every whole number greater than 1 has exactly one prime factorization, no matter how you build the tree.

Try this together

Two or three minutes at the table is enough.

1. Open Factor tree. The number is 60. Ask your kid to choose the first split. Any split is allowed.
2. Keep splitting together until every leaf is prime. Read the factorization at the bottom.
3. Change the number to 97. Ask what happens. It is already prime, so there is nothing to split.
4. Switch to GCF & LCM and try 12 and 18. Point at the shared primes, then read the GCF and the LCM.

If your kid's tree looks different from a friend's, that is expected. The leaves come out the same.

Words you might hear

• Prime: a whole number greater than 1 whose only factors are 1 and itself.
• Composite: a number with more than two factors, like 12.
• Factor tree: a diagram that splits a number into factors until only primes remain.
• Prime factorization: the list of primes that multiply to the number, like 60 = 2 × 2 × 3 × 5.

The prime factorization of a number is unique. Two different trees for one number always agree at the end.

Deeper dive

Start with 60 and split it into 6 × 10. Split 6 into 2 × 3 and 10 into 2 × 5. Every leaf is now prime, so 60 = 2 × 2 × 3 × 5.

If you start the same number as 2 × 30, you still finish with 2 × 2 × 3 × 5. The shared primes give the GCF and the highest powers give the LCM.