Decide whether each of the following statements is true or f

Decide whether each of the following statements is true or false, and give a proof or counterexample: (a) gcd(a, b) = gcd(a, c) gcd(a^2, b^2) = gcd (a^2, c^2). (b) gcd (a, = gcd (a, gcd (a, = gcd (a, b, c). (c) Suppose a, b, c are all non-zero. If p| (a^2 + b^2) and p | (b^2 + c^2), then p| (a^2 + c^2).

Solution

(a) Let gcd(a,b) = gcd(a,c) = k

Therefore, k=0(moda), k=0(modb), k=0(modc)

squaring these equations, we get:

k²=0(moda²), k²=0(modb²), k²=0(modc²)

Therefore, gcd(a²,b²) = gcd(a²,c²) = k²

Hence, this statement is true.

(b) This statement is true as well. It can be proved the same way we proved the previous one.

(c) This statement is false. let us see a counter example.

Say, a=3, b=4 and c=2 and p = 5

For these numbers, 5 does not divide (3²+2²) = 13