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Definition: Let p be a prime and 0 < n then the p-exponent of n, denoted...

Definition: Let p be a prime and 0 < n then the p-exponent of n, denoted ε(n, p) is the largest number k such that pk | n.

Note: for p does not divide n we have ε(n,p) = 0

Notation: Let n ∈ N+ we denote the set {p : p is prime and p | n} by Pr(n). Observe that Pr(n) ⊆ {2, 3, . . . n} so that Pr(n) is finite.

Problem: Let a, b be positive integers. Show that a | b if and only if Pr(a) ⊆ Pr(b) and for all p ∈ Pr(a), ε(a,p) ≤ ε(b,p).

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i will solved by using definition and in case 2 nd i was taken p1 equal to q1 which only for understanding perpose

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