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Number Theory (MA3Z7)
Problem Sheet II
1. Let a, b ∈ N. Prove that if there exist integers m, n such that ma + nb = 1, then (a, b) = 1.
2. Prove that if (a, b) = 1, then (a n , bk ) = 1 for all n, k ∈ N.
[Hint: Use Q4 from Problems I]
3. (a) Show that if (a, b) = 1 and (b/a)m ∈ N, then b = 1.
(b) Deduce that if n is not the mth power of a positive integer, then m√n is irrational.
4. (Divisibility criterion for 11.) Prove that a number akak−1 . . . a1a0 (as written in base 10) is divisible by 11 if and only if
a0 − a1 + a2 − · · · + (−1)k ak is divisible by 11.
5. (Alternative proof of Theorem 2.6.) Use Theorem 1.3 to prove that the congruence ax ≡ b (mod m) has a solution whenever (a, m) = 1.
Show further that it is unique modulo m.