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Q: When a number is a multiple of another the greatest common factor of the numbers is the greater of the numbers?

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No, the lesser.

There is no greatest common multiples for whatever common multiple is claimed to be the greatest the lowest common multiple of the numbers (in this case 15) can be added to get an even greater common multiple.

In that scenario, the GCF is the lesser of the numbers. The LCM is the greater.

The GCF of the numbers is the greatest common factor no matter what their relationship is. When one number is a multiple of another number, the GCF is the smaller number.

There can never be a greatest common multiple of one number for two reasons:"Common" refers to a multiple that is common to two or more numbers. You cannot have a multiple that is common, but only to one number.If X is the greatest common multiple of a set of numbers, then any multiple of X will also be a common multiple of each member of the set and it will be greater than X. And then, any multiple of this number will be a multiple of each member of the set and will be greater still. And then ...

The answer is sometimes - when the multiple in question is 1.

There can be no such number. Suppose x is the least common multiple (LCM) of these numbers. Then 2*LCM is also a common multiple and is greater. And then 3*LCM is a common multiple and greater still. And so on, for ever.

There is no greatest common multiple of any integers as whatever number is said to be it, the lowest common multiple of the numbers can be added to get an even greater common multiple. If you mean least common multiple (the lowest (positive) integer that can be divided by the numbers without a remainder), the answer is 312. If you mean the greatest common factor (the greatest (positive) integer that can divide into the numbers without any remainder). the answer is 4.

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

The greatest common multiple of any set of numbers is infinite. The greatest common multiple of any set of numbers will never be one.

The greatest common multiple of any two numbers is infinite.

There is no "greatest common multiple" of any two numbers. Whatever their product is, it can be multiplied by any positive integer to yield an even greater number that is also a multiple of the first two. Thus, the number of multiples is infinite.

The greatest common multiple of any two numbers is infinite.

Never. The greatest common multiple of any two numbers is infinite.

There can be no greatest denominator. Suppose x is the greatest denominator of two numbers a and b. That is to say, x is a multiple of a, x is a multiple of b and x is the greatest such number. Now consider the number 2x. It is a multiple of a, it is a multiple of b and it is greater than x. This contradicts the statement that x is the greatest such number. So such an x cannot exist.

There are no such numbers because there is really no such thing as a "greatest common multiple". If the numbers have 5 as a common multiple then 10 will also be a common multiple and clearly, 10 is greater than 5. So 5 cannot be the greatest common multiple. In fact, once you find the least common multiple of a set of numbers, you can keep adding the LCM to itself over and over again. Each new number you get will be a common multiple of your set of numbers, but each new number will always be larger than the previous. This means that you can keep adding while the number approaches infinity and you will still never find a greatest multiple.

The greatest common multiple is infinite; numbers don't stop.

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

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

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

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

Since the list of numbers is infinite, there is no such thing as the Greatest Common Multiple.

There is no such thing as a greatest common multiple for any numbers. There is a greatest common divisor and that is 2.

There is no greatest common multiple of these numbers... or indeed of any pair of numbers. Once you get a common multiple, you can multiply that by any integer, as large as you want, to get a larger common multiple.

The definition of the greatest common multiple of any two numbers is an infinite number.