Reference no: EM132323140

Assignment - Solve these problems:

1. For an arbitrary integer a, verify the following:

(a) 2|a(a + 1), and 3|a(a + 1)(a + 2).

(b) 3|a(2a² + 7).

(c) If a is odd, then 32|(a² + 3)(a² + 7).

2. Prove that if a and b are both odd integers, then 16|a⁴ + b⁴ - 2.

3. Prove the following:

(a) The sum of the squares of two odd integers cannot be a perfect square.

(b) The product of four consecutive integers is 1 less than a perfect square.

4. Establish that the difference of two consecutive cubes is never divisible by 2.

5. For a nonzero integer a, show that gcd(a, 0) = |a|, gcd(a, a) = |a|, and gcd(a, 1) = 1.

6. If a and b are integers, not both of which are zero, verify that

gcd(a, b) = gcd(-a, b) = gcd(a, -b) = gcd( -a, -b)