Unique factorization

For any prime p and any positive integer n, denote the highest power of p

dividing n by Op.n/. That is, Corollary 2.11 guarantees

that Op is well-defined.

Op.n/ D e;

where pe j n but peC1 − n. If m and n are positive integers, prove that

(i) Op.mn/ D Op.m/ C Op.n/

(ii) Op.m C n/  min

°

Op.m/;Op.n/

.When does equality occur?

There is a generalization of Exercise 1.6 on page 6. Using a (tricky) inductive

proof (see FCAA [26], p. 11), we can prove the Inequality of the Means: if n  2

and a1; : : : ; an are positive numbers, then

npa1


an  1

n .a1 C


Can/: