In a classroom exam where each question is worth 1 point and the number of exam questions are unknown. If the student receives results of a test in a percentage with decimals, let's say 78.5%, what is the formula to determine the smallest integer dividend that is the number of questions on the test?
For example:
- 78.5% = 785/1000
- 78.5% = 471/600
- 78.5% = 157/200
Through successive approximation, 200 appears to be the number of questions on the test, because 157 out of 200 yields the given percentage.
Given an input of $x$ percentage, where $x$ can be represented on base $10$ as $m$ digits before the decimal point and $n$ digits after the decimal point, the minimum number of questions is:
$$\frac{10^{m+n}}{\gcd(10^{m+n},10^nx)}$$