To prove the statement by contrapositive, we must first understand the structure of the statement. The original statement is:

"If $10a \equiv 10b \pmod{n}$, then $n \mid 10$."

The contrapositive of this statement is:

"If $n \nmid 10$, then $10a \not\equiv 10b \pmod{n}$."

Step-by-Step Proof by Contrapositive:

1. Assume $n \nmid 10$:

This means that $n$ does not divide 10. In other words, 10 is not divisible by $n$, which implies that the greatest common divisor $\gcd(n, 10) > 1$.

2. We need to prove that $10a \not\equiv 10b \pmod{n}$:

This means we need to show that under this assumption, there are integers $a$ and $b$ such that $10a \not\equiv 10b \pmod{n}$. In other words, the difference $10a - 10b$ is not divisible by $n$.

3. Derive the equation from the assumption $n \nmid 10$:

Since $n \nmid 10$, it follows that 10 is not divisible by $n$. Therefore, if we multiply any integers $a$ and $b$ by 10, the products $10a$ and $10b$ will still not behave in a way that makes (