What does the term greatest common multiple mean?

Answer 1

The term Greatest Common Multiple has no meaning; there is no such thing as a greatest multiple.

There are the terms:

Lowest Common Multiple (LCM) which is the smallest positive (whole) number which is a multiple of two, or more, numbers. eg LCM(4, 10) = 20 as 20 is the first number greater than 0 which is a multiple of both 4 and 10.

Highest Common Factor (HCF), or Greatest Common Factor (GCF) which is the largest positive (whole) number that divides into all the numbers (without remainder); this can also said to be the largest positive number (whole) number which has all the numbers as some multiple of it. eg HCF(25, 60) = 5 since 5 is the highest positive (whole) number which divides into 25 and 60 (without remainder).

The Lowest Common Factor of any set of numbers is 1 as 1 is the smallest positive (whole) number and it divides into every other (whole) number without remainder; this is not very useful.

Answer 2

There is no such thing. A common multiple (of two or more numbers) is a number that is a multiple of all of them. There is no GREATEST common multiple. For example, 24 is a common multiple of 4 and 6. But if you multiply 24 x 2 - and you can multiply the result by 2, over and over again - you can always get a greater common multiple.