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

1. Let n be a positive integer, . 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 a^d – 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:
• is a cube;
• is a square.
5. Find the remainder when is divided by 3, where [x] denotes the largest integer ≤ x. E.g. [3.5] = [3] = 3 and [-4.5] = [-5] = -5.