What is the GCD for 60 and 25?

Answer 1

There cannot be a greatest common denominator. For suppose X were the greatest common denominator. That means X is a multiple of 60 and 25 and that X is the greatest such number.

Consider 2X.

If X is a multiple of 60 and 25 then 2X will certainly be a multiple of the two numbers and 2X will also be greater than X. But that contradicts the proposition that X is the greatest such number.

That contradiction shows that there cannot be a GCD for 60 and 25. The argument can easily be generalised to prove that there cannot be a GCD for any set of numbers.