The homework for last week was a little harder than the prior one:
- Let n be a positive integer, . Prove that the sum of the divisors of n is a multiple of 24.
- Let N = 210 × 39 × 58. Calculate:
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).
Prove that there exist distinct positive integers x1, x2, …, x2011, such that both conditions hold:
- the number of divisors of N;
- the sum of all divisors of N;
- the number of i modulo N which are coprime to N.
Find the remainder when is divided by 3, where [x] denotes the largest integer ≤ x. E.g. [3.5] =  = 3 and [-4.5] = [-5] = -5.
- is a cube;
- is a square.
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