A Story of an Algorithm
Consider an algorithm for positive integers \(n\) and \(k\)-
Take any multiple of \(n\). Multiply the last digit by \(k\) and then, add the resulting number to the remaining number to get a number \(a\).
For Example- For \(n=7\), \(k=3\) and multiple of \(7=105\), the algorithm would give you \(a=10+5×3\).
How many values of \(n <1000\) are there such that there exists at least one \(k\) for which \(a\) is always a multiple of \(n\)?