Homework (29 Oct 2011)

The homework for last week was a little harder than the prior one:

  1. Let n be a positive integer, n \equiv 23 \pmod {24}. Prove that the sum of the divisors of n is a multiple of 24.
  2. Let N = 210 × 39 × 58. Calculate:
    • the number of divisors of N;
    • the sum of all divisors of N;
    • the number of i modulo N which are coprime to N.
  3. Let p be a prime number and a be an integer not divisible by p. Suppose d is the smallest positive integer for which ad – 1 is divisible by p. Prove that d | (p-1).
  4. Prove that there exist distinct positive integers x1, x2, …, x2011, such that both conditions hold:
    • x_1^2 + x_2^2 + \dots + x_{2011}^2 is a cube;
    • x_1^3 + x_2^3 + \dots + x_{2011}^3 is a square.
  5. Find the remainder when [(1 + \sqrt 2)^{2011}] is divided by 3, where [x] denotes the largest integer ≤ x. E.g. [3.5] = [3] = 3 and [-4.5] = [-5] = -5.
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