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Sum of Remainders

Source: 2021 Fall AMC 12B 25

November 17, 2021
AMCAMC 12AMC 12 B2021 AMC Fallnumber theorycombinatoricsCasework

Problem Statement

For nn a positive integer, let R(n)R(n) be the sum of the remainders when nn is divided by 22, 33, 44, 55, 66, 77, 88, 99, and 1010. For example, R(15)=1+0+3+0+3+1+7+6+5=26R(15) = 1+0+3+0+3+1+7+6+5=26. How many two-digit positive integers nn satisfy R(n)=R(n+1) ?R(n) = R(n+1)\,?
<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>3<spanclass=′latex−bold′>(E)</span>4<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>2\qquad<span class='latex-bold'>(D) </span>3\qquad<span class='latex-bold'>(E) </span>4
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