Question

1. Prove that an integer a is divisible by 5 if and only if a2 is divisible by 5.

2. Deduce that 98765432 is not a perfect square. Hint: You can use any theorem/proposition or whatever was proved in class.

3. Prove that for all integers n,a,b and c, if n | (a−b) and n | (b−c) then n | (a−c).

4. Prove that for any two consecutive integers, n and n + 1 we have that gcd(n,n + 1) = 1. Hint: Prove that for all integers k,a and b, if k | a and k | b then k | (a−b).

5. Use the Euclidean Algorithm to find the gcd(a,b) in the cases:

1* a = 1072 and b = 462,

2*   a = 888 and b = 54.

6. Prove that the sum of four consecutive natural numbers is never divisible by 4.

Note: please answer Q# 1,2,3,4,5 and 6 with step by step, so i can follow up with you and understand.

Answer 1