2009 UNCO Math Contest II Problems/Problem 2
Problem
(a) Let . For how many between and inclusive is a multiple of ?
(b) For how many between and inclusive is a multiple of 5?
Solution
(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 .
See also
2009 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |