Which is the smallest integer number $a$ so that the highest common factor of $a$ and $a+5$ is not $1$ and the highest common factor of $a$ and $a+7$ is not $1$ either?
I think that it is $35$. Am I right ?
2026-04-02 17:51:05.1775152265
Smallest $a$ such that both $a$ and $a+5$ and $a$ and $a+7$ have a common factor
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Since the highest common factor of $a$ and $a+5$ is the same as that of $a$ and $(a+5) - a = 5$, one needs $5 \mid a$ for it not to be $1$. Likewise you need $7 \mid a$.
So, $5$ and $7$ divide $a$, and the result you claimed follows in one more step.
(This assumes that you are looking for a positive solution. If you allows negative numbers there is no smallest one, as any multiple of $35$ would work. So $-35$, $-350$, $-35000000000000000$ and so on. )