Question

Let a be prime and b be a positive integer. Prove/disprove, that if a divides b^2 then a divides b.

Answer #1

Let a positive integer n be called a super exponential number if
its prime factorization contains at least one prime to a power of
1000 or larger. Prove or disprove the following statement: There
exist two consecutive super exponential numbers.

Let
h € N be a prime.
Now prove for all b € N, h divides b^h -b.

Suppose p is a positive prime integer and k is an integer
satisfying 1 ≤ k ≤ p − 1. Prove that p divides p!/ (k! (p-k)!).

Prove that the divides relation is a partial order on the set of
positive integer.

Prove that the divides relation is a partial order on the set of
positive integer.

Prove or disprove that 3|(n^3 − n) for every positive integer
n.

(§2.1) Let a,b,p,n ∈Z with n > 1.
(a) Prove or disprove: If ab ≡ 0 (mod n), then a ≡ 0 (mod n) or
b ≡ 0 (mod n).
(b) Prove or disprove: Suppose p is a positive prime. If ab ≡ 0
(mod p), then a ≡ 0 (mod p) or b ≡ 0 (mod p).

a,b,c are positive integers
if a divides a + b and b divides b+c prove a divides a+c

(a) Prove or disprove the statement (where n is an integer): If
3n + 2 is even, then n is even.
(b) Prove or disprove the statement: For irrational numbers x
and y, the product xy is irrational.

Prove that for any integer a, k and prime p, the following three
statements are all equivalent: p divides a, p divides a^k, and p^k
divides a^k.

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