(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then
a∣c.
(B) Let p ≥ 2. Prove that if 2p−1 is prime, then p
must also be prime.
(Abstract Algebra)
Prove that there are domains R containing a pair of elements
havig no gcd.(Hint: let R be a subring of F[x], for a field F,
where R consistes of all polynomials with no linear terms, then
show that x5 and x6 have no gcd) .
Suppose a > b are natural numbers such that gcd(a, b) =
1.
Compute each quantity below, or explain why it cannot be
determined (i.e. more than one value is possible).
(a) gcd(a3, b2)
(b) gcd(a + b, 2a + 3b)
(c) gcd(2a,4b)
(1)Prove that for every a, b ∈ R, |a + b| = |a| + |b| ⇐⇒ ab ≥ 0.
Hint: Write |a + b| 2 = (|a| + |b|) 2 and expand.
(2) Prove that for every x, y, z ∈ R, |x − z| = |x − y| + |y −
z| ⇐⇒ (x ≤ y ≤ z or z ≤ y ≤ x). Hint: Use part (1) to prove part
(2).