Question

Prove that if a|n and b|n and gcd(a,b) = 1 then ab|n.

Answer #1

prove that if gcd(a,b)=1 then gcd (a-b,a+b,ab)=1

Given that the gcd(a, m) =1 and gcd(b, m) = 1. Prove that gcd(ab,
m) =1

4. Let a, b, c be integers.
(a) Prove if gcd(ab, c) = 1, then gcd(a, c) = 1 and gcd(b, c) =
1. (Hint: use the GCD characterization theorem.)
(b) Prove if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) =
1. (Hint: you can use the GCD characterization theorem again but
you may need to multiply equations.)
(c) You have now proved that “gcd(a, c) = 1 and gcd(b, c) = 1 if
and...

Prove that for positive integers a and b,
gcd(a,b)lcm(a,b) = ab. There are nice proofs that do not use the
prime factorizations of a and b.

Prove that if gcd(a,b)=1 and c|(a+b), then
gcd(a,c)=gcd(b,c)=1.

Let |a| = n. Prove that <a^i> = <a^j> if and only if
gcd(n,i) = gcd (n,j) and |a^i| = |a^j| if and only if gcd(n,i) =
gcd(n,j).

Assume that gcd(a, m) = 1, gcd(a, n) = 1, and gcd(m, n) = 1.
Assume that a has order s modulo m and order t modulo n. What is
the order of a modulo mn? Prove that your answer is correct

(§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) Let N be an even integer, prove that GCD (N + 2, N) = 2.
(b) What’s the GCD (N + 2, N) if N is an odd integer?

Prove that if a > b then gcd(a, b) = gcd(b, a mod b).

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