Mathematical number theory proof problem

Question

If number n is divisible by a and n = p₁^z1 p₂^z2 ... p_k^zk, prove that n = p₁^l1 p₂^l2 ... p_k^lk with l_i ≤ z_i with all i = 1,...,k

Answer 1

n is divisible by a, so n=ka where k is an integer.

a and k must both be products of the same primes making up n.

(If the decomposition of a into primes included a prime p not included in n, then p would divide a but not divide n so n would not be divisible by a because n=ka would be impossible)

So a = p₁^a1 p₂^a2 ...

k = p₁^k1p₂^k2 ......

a1, a2 .... k1, k2 >=0, are integers

So by definition a1 + k1 = z1, a2 + k2 = z2 ....

So a1 <= z1, a2 <=z2, ....