What is the smallest possible common multiple of two distinct integers, each greater than 251?

Incorrect.
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This is indeed a possible common multiple of two distinct integers that are greater than 251, but this is not the *least* possible common multiple of two such numbers.

Incorrect.
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Even if you missed out on the fact that the numbers have to be distinct, and therefore chose 252 twice, the LCM of 252 and 252 is simply 252, and not $$ 252^2$$.

Incorrect.
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Incorrect.
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The numbers have to be distinct (different from one another), and thus they cannot both be 252.

Correct.
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The general principle behind this question is that the smallest possible LCM of two integers is reached when the two integers have *the greatest number of factors in common* (which are only counted once in the LCM). This should lead you to use 252 (the smallest possible integer) and a multiple of 252 so that the two integers will have the greatest number of common factors that are only counted once in the LCM.
Thus, you can use $$252$$ and $$(252\times 2)$$ so that the common multiple will be the greater of the two. Since $$(2\times 252)$$ is divisible by both itself and by 252, it is the LCM of 252 and itself.
>$$252\times 2 = 504$$
Therefore, 504 is the correct answer.

$$252$$

$$504$$

$$1{,}506$$

$$252^2$$

$$252\times 253$$