2009 UNCO Math Contest II Problems/Problem 2
(a) Let . For how many between and inclusive is a multiple of ?
(b) For how many between and inclusive is a multiple of 5?
(a) Looking at the units digits, we need the units digit of to be either or . We know that will always have a units digit of . Looking at , however, cycles every four powers with units digits and in that order. We see that we can only get a units digit of if we have as a units digit for , and there is no way to get as a units digit. Therefore, our answer is because the four units digits cycle times in the integers to .
(b) Similarly, cycles every four powers with units digits and in that order. And cycles every two powers with units digits and . Together the units digit of their sum is for Mod() , and for Mod(). So the answer is .
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