代做Number Theory (MA3Z7) Problem Sheet II代做Statistics统计

日期：2024-08-09 12:50

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.