🌳 Prime Factor Tree

Every number has a tree hiding inside it! You split the number into factors, branch by branch, and at the tips of the branches the prime numbers bloom. Let’s grow the tree of 60 together!

A prime number is a number greater than 1 that divides only by 1 and itself. 6 dots can make 2 rows of 3, but 7 dots can only make a single row. So 7 is prime and 6 is not.

Careful: 1 is not prime, because it has only one factor. 2 is the smallest prime and the only even prime. Other even numbers like 4, 6 and 8 divide by 2, so they are not prime.

Quick tests help before you grow the tree. If the last digit is even, the number divides by 2. If the digit sum is a multiple of 3, it divides by 3. If the last digit is 0 or 5, it divides by 5. 60 passes all three tests!

Split 60 into two factors: 60 = 6 × 10. Then 6 = 2 × 3 and 10 = 2 × 5. When every branch tip is prime, the tree blooms. This is a prime factor tree.

A common mistake: starting with 60 = 4 × 15 and stopping at 4. But 4 is not prime! Keep branching: 4 = 2 × 2. The tree is only finished when every tip is prime.

Two gardeners split 60 with different branches: one started with 6 × 10, the other with 2 × 30. The trees look different, but the flowers are the same: 2, 2, 3 and 5. Whichever branch you start with, the prime factors come out the same.

Let’s write the flowers as a product: 60 = 2 × 2 × 3 × 5. Writing a number as a product of primes like this is called prime factorisation. You can also write it with a power: 2² × 3 × 5. The order can change, but the numbers are always the same.

The tree of 42: 42 = 2 × 3 × 7. Both trees have a 2 flower and a 3 flower, so 2, 3 and 2 × 3 = 6 divide both numbers. 1 divides every number too. The common factors of 60 and 42 are 1, 2, 3 and 6.

Summary: a prime has exactly two factors, and 1 is not prime. Grow the tree until every branch tip is prime: 60 = 2 × 2 × 3 × 5. The common flowers of two trees help you find the common factors. Now try it in Prime Forest!