Let $s$ be the set of all numbers of the form $abc$ where $ (a, b, c) $ is a pythagorean triple. find the gcd of all elements of $s$.

S = {a, b, c}

Here a, b, c are Pythagorean triplets.

Let us first take the smallest triplet 3,4,5

Now as 3 is a prime number, GCD shall be 1

Another one is 5, 12, 13.

Here 13 is a prime number, so GCD shall be 1

Next one is 20,21, 29

Here 29 is a prime number, so GCD = 1

For 15, 8, 17

17 is a prime number so GCD = 1

Now when none of them is a prime number'

For 77,36, 85

77 = 7 * 11

36 = 2 * 2 * 3 * 3

85 = 5 * 17

Here also the GCD is 1

So For S = {a, b, c} the Pythagorean triplets the GCD is always 1.