# The least common multiple of a number "n" and 6 is 24. What is the value for "n"?

## Expert Answers Let's consider the prime factorization of both given numbers, `6` and `24.`

It is clear that `6 = 2^1 * 3^1`  and  `24 = 2^3 * 3^1.`

Hence the number `n` must contain `2` exactly in degree `3` in its prime factorization. If it would have `2` in greater degree, the LCM of `6` and `n` would have `2` in that greater degree, and if in less, then in less.

Also `n` may contain `3` in degree not greater than `1.` It may contain `3` in degrees `0` or `1,` because `6` already have `3^1` and `24` also.

And it cannot have any other prime factors.

This gives us two options for `n:`  `2^3 * 3^0 = 8` and `2^3 * 3^1 = 24.`

Approved by eNotes Editorial Team

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