OIMU 2008 Problem 1

Let $n$ be a positive integer that is not divisible by either $2$ or $5$ . In the decimal expansion of $\frac{1}{n}= 0.a_1a_2a_3\cdots$ a finite number of digits after the decimal point are chosen arbitrarily to be deleted. Clearly the decimal number obtained by this procedure is also rational, so it's equal to $\frac{a}{b}$ for some integers $a,b$ . Prove that $b$ is divisible by $n$ .