1. Suppose n ≡ 7 (mod 8). Show that n ≠ x2 + y2 + z2 for any x, y, z ε Z.
2. Prove ∀n ε Z, that n is divisible by 9 if and only if the sum of its digits is divisible by 9.
3. Prove that it is always possible to make postage of exactly n cents for all n ≥ 32 using only 5 and 9 cent stamps.
4. Prove that every fourth Fibonacci number is a multiple of 3.
In other words, show that 3 | f4n ∀n ≥ 1.
5. Let bn be the sequence recursively defined by b0 = 1, b1 = 5 and, for n > 1,
bn = b[n/3]+2b[n/3]
(a) Compute b26 and b27.
(b) Guess a formula for bn when n = 3t for t ≥ 0 and then use mathematical induction to prove that your guess is correct. (Be sure to include a careful statement of what you are trying to prove).