Prove that it is NOT true that for every integer $n$, 60 divides $n$ if and only if 6 divides $n$ and 10 divides $n$.

Question

This is Velleman's exercise 3.4.26 (b):

Prove that it is NOT true that for every integer $n$, 60 divides $n$ iff 6 divides $n$ and 10 divides $n$.

I do understand that a number will be divisible by 6 and 10 if it is divisible by 60 and that it will not necessarily be divisible by 60 if it is divisible by 6 and by 10. 30 is an example of it.

I still have an issue actually discovering and writing up the proof. To illustrate the issue and to put the question in context, I would like to refer to the first part of the question already asked by another user, Velleman's exercise 3.4.26 (a). Thanks in advance.

Answer 1

Consider the number $30$. It is divisible by $6$ and $10$ and not by $60$. We have found a counterexample and therefore the statement is not true.

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The statement $ab$ divides $n$ if and only if $a$ divides $n$ and $b$ divides $n$ is only true if $a$ and $b$ are relatively prime (share no prime factors).

Otherwise we can take the least common multiple of $a$ and $b$, this number will be smaller than $ab$ (since they are not relatively prime) and it will be divisible by $a$ and $b$. The least common multiple will always be a counterexample if $a$ and $b$ are not relatively prime.