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.