1: (a) What are the last two digits of 3²⁰¹⁴?

(b) What are the last two digits of (97)²⁰⁰³?

2: (a) Prove that this sequence contains no squares:

11, 111, 1111, 11111, ...

(b) Prove that if 2n + 1 and 3n + 1 are both perfect squares, then n is divisible by 40.

3: (a) Prove that any subset of 55 numbers chosen from {1, 2, 3, ..., 100} must contain two numbers differing by 9.

(b) Prove that for any set of n integers, some subset of them has sum is divisible by n.

4: For a positive integer n, define the digital sum to be the number you get by summing all its digits, then summing all the digits of that number, and repeating until you get a single digit number. Prove that if you take any pair of twin primes (two consecutive odd numbers that are both prime) other than 3 and 5, then the digital sum of their product is 8.

5: The number d₁d₂ · · · d₉ has nine (not necessarily distinct) decimal digits. The number e₁e₂ · · · e₉ is such that each of the nine 9-digit numbers formed by replacing just one of the digits d_i in d₁d₂ · · · d₉ by the corresponding digit e_i is divisible by 7. The number f₁f₂· · · f₉ is related to e₁e₂ · · · e₉ in the same way: that is, each of the nine 9-digit numbers formed by replacing just one of the digits e_i by the corresponding f_i is divisible by 7.

Show that, for each i, d_i - f_i is divisible by 7.

6: A classy social security number? Use the digits 1 through 9 once only to form a nine-digit number such that the first (leftmost) 8 digits form a number divisible by 8, the first 7 digits form a number divisible by 7, and so on. How many such numbers are there?