Factor Trees

Every composite number can be broken down into primes. Take 90 and split it into any factor pair, say 9 × 10. Neither 9 nor 10 is prime, so split again: 9 is 3 × 3 and 10 is 2 × 5. Now every piece is prime and the splitting stops. Drawn out, the pieces hang below 90 like branches, so this is a factor tree, and the primes at the bottom are its leaves.

90 = 2 × 3 × 3 × 5

The leaves multiply back to 90, because every split kept the value the same. Written smallest first, they are the prime factorization of 90.

That also finds a leaf that has gone missing. Say a tree for 70 shows the leaves 2 and 5 and one more: 2 × 5 × n = 70 (the board writes 5 times n as 5n). The leaves you have make 2 × 5, which is 10, and 70 / 10 is 7, a prime. On the board you write the claim n = 7 and press Check.