## Saturday, June 20, 2015

### Problems conceived by me

1. Recall the laws of divisibility by 2-11. Prove them.
2. Devise laws of remainders like "Remainder of a number by 2-11" etc...
a. [Open problem] Devise law of remainder for 7.
3. Prove Fermat's little theorem,
a^p = a (mod p) where p is a prime number.
4. For a number N = \pi_{i=1}^k p_i^{e_i}, given the prime number expansion, find out the sum of all factors of N (including 1 and N itself). Prove that it is,
\pi_{i=1}^k [(p_i^{e_i + 1} - 1)/(p_i - 1)]
5. Sum of reciprocals of all factors of a number is given by,
\pi_{i=1}^k [(p_i - p_i^{-e_i})/(p_i - 1)].
6. Prove that for a number N = \pi_{i=1}^k p_i^{e_i}, the number of factors is given by
\pi_{i=1}^k (e_i + 1).
7. Only numbers having odd number of factors are ____________ numbers.
8.