Problem:
An integer is selected at random in the range . What is the probability that the remainder when is divided by is ?
Answer Choices:
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Solution:
An integer will have a remainder of when divided by if and only if the units digit is either or . The randomly selected positive integer will itself have a units digit of each of the numbers from through with equal probability. This digit of alone will determine the units digit of . Computing the th power of each of these digits by squaring the units digit four times yields one , one , four s, and four s. The probability is therefore .
Note: This result also follows from Fermat's Little Theorem.
The problems on this page are the property of the MAA's American Mathematics Competitions