Problem:
What is the remainder when 72024+72025+72026 is divided by 19 ?
Answer Choices:
A. 0
B. 1
C. 7
D. 11
E. 18
Solution:
The quantity in question is seen to be a multiple of 19 as follows:
72024+72025+72026=72024(1+7+72)=72024â‹…57=72024â‹…3â‹…19
Therefore the remainder when it is divided by 19 is (A)0​.
OR
Working modulo 19,70=1,71=7⋅1=7,72=7⋅7=49≡11, and 73≡7⋅11=77≡1. 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 1+7+11=19, so the remainder when the given sum is divided by 19 is (A)0​.
The problems on this page are the property of the MAA's American Mathematics Competitions