A Story of an Algorithm

Consider an algorithm for positive integers nn and kk-

Take any multiple of nn. Multiply the last digit by kk and then, add the resulting number to the remaining number to get a number aa.

For Example- For n=7n=7, k=3k=3 and multiple of 7=1057=105, the algorithm would give you a=10+5×3a=10+5×3.

How many values of n<1000n <1000 are there such that there exists at least one kk for which aa is always a multiple of nn?

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