2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1
What are the last two digits of ?
In this problem, we will use the Chinese remainder theorem. This question is asking us to find . By the Chinese remainder theorem, we can find and , and then "combine" them. . To find , we'll look for a pattern. The pattern is . Since there are 20 terms, and , we have that . Now, we want to find a number that's and . We find this number by clever guess and check. We know our number is odd, so it is . First, we try , but that doesn't work. Next, we try , and that works! So, our answer is
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