Answer to Question #81580 in Abstract Algebra for Rozina

Question

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

Answer 1

If p prime and p|ab then p|a or p|b.
Let's prove, that if p|a^50, then p|a.
Actually, if p|a^50, then p|a(a^49). That means, that either p|a, either p|a^49, and then a|a(a^48) etc.
Finally we get p|a^2, so p|aa, so p|a in both cases.
Now, we know that p|a, so a=pb, and a^50=(pb)^50=(p^50)(b^50), so (p^50)|(a^50).