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There can never be a greatest common multiple of one number for two reasons:

  1. "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.
  2. 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 ...
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13y ago

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

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8y ago
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