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 di in d1d2 · · · d9 by the corresponding digit ei 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 ei by the corresponding fi is divisible by 7.
Show that, for each i, di - fi 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?