What is the greatest common multiple of 100 and 140?

Answer 1

No two numbers can have a greatest common multiple.

For, suppose x is the greatest common multiple of two numbers, a and b.

That means x = m*a and x = n*b where m and n are some positive integers.

Then any multiple of x, say p*x where p is an integer, will be a multiple of a and b because

p*x = p*(m*a) = (p*m)*a

p*y = p*(n*b) = (p*n)*b

m,n,p are integers so p*m and p*n are integers and p*x > x

So p*x is a common multiple, and is greater than x, contradicting the assumption that x is the greatest.

Answer 2

The greatest common multiple of any set of integers is infinite.