Using Exponents

A tree for 80 ends with the leaves 2, 2, 2, 2 and 5. Writing the same prime four times is long, so use a power, as in Section 1: 2 × 2 × 2 × 2 is 2^4. The prime factorization of 80 is then 2^4 × 5. The exponent counts how many times that prime appears as a leaf.

A prime that appears once gets no exponent, like the 5 here. Written this way, a factorization reads at a glance: 2^4 × 5 says straight away that 80 holds four 2s and one 5.

Going the other way finds a missing exponent. If 2^n × 5 = 80, the 2s together must make 80 / 5, which is 16. Now count how many 2s multiply to 16: 2, 4, 8, 16 takes four of them, so the claim is n = 4.