Problem:
What is the remainder when is divided by 19 ?
Answer Choices:
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Solution:
The quantity in question is seen to be a multiple of 19 as follows:
Therefore the remainder when it is divided by 19 is .
Working modulo , and . Therefore the remainders when dividing successive nonegative powers of 7 by 19 repeat with period 3 . The sum of any three consecutive remainders is therefore , so the remainder when the given sum is divided by 19 is .
The problems on this page are the property of the MAA's American Mathematics Competitions