Easy is in the eyes of the doer. It really depends on what one prefers.

Personally I prefer to factor numbers into their prime factor form.

For example 144000=(2^7)(3^2)(5^3) and 68600=(2^3)(5^2)(7^3).

So gcd(144000, 68600 )=(2^3)(5^2).

Use the common primes with their least powers.

This method works with more than two numbers.