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* = 2^{10} × 3^{9} × 5^{8}. 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*.

- 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).
- Prove that there exist distinct positive integers
*x*_{1}, *x*_{2}, …, *x*_{2011}, such that both conditions hold:
- is a cube;
- is a square.

- 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.

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