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.

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.

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