1 suppose n equiv 7 mod 8 show that n ne x2 y2

Question

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 | f_4n ∀n  ≥ 1.

5. Let b_n 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 = 3^t 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).