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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.

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Q: What is the greatest denominator for 4 and 19?
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