2017 AMC 10B Problems/Problem 14
Problem
An integer is selected at random in the range . What is the probability that the remainder when is divided by is ?
Solution 1
By [url = Fermat's Little Theorem]https://artofproblemsolving.com/wiki/index.php?title=Fermat%27s_Little_Theorem[/url], when N is relatively prime to 5. However, this happens with probability .
Solution 2
Note that the patterns for the units digits repeat, so in a sense we only need to find the patterns for the digits . The pattern for is , no matter what power, so doesn't work. Likewise, the pattern for is always . Doing the same for the rest of the digits, we find that the units digits of , ,, ,, , and all have the remainder of when divided by , so .
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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