Question

Let p be a prime and a∈N such that p|a^50.Show that p^50|a^50.?

Answer 1

See below.

Explanation:

If `p` is prime and `a in NN` is such that `p | a^50` with `a= prod_k f_k^(alpha_k)`with `f_k` being the prime factors for `a`,

then

`a^50 = prod_k f_k^(50 alpha_k)` then if `p` is prime one of the `f_k` must be equal to `p` so `f_(k_0) = p` and `a^50` has a factor which is `f_(k_0)^(50 alpha_(k_0)) = p^(50alpha_(k_0))`

then `p^50|a^50`