A Story of an Algorithm

Number Theory Level 5

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 \(n\)?

Like this one? This is also nice.

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