2019 AMC 10C Problems/Problem 21

Problem

Jack eats candy while abiding to the following rule: On Day $n$, he eats exactly $n^3$ pieces of candy if $n$ is odd and exactly $n^2$ pieces of candy if $n$ is even. Let $N$ be the number of candies Jack has eaten after Day 2019. Find the last two digits of $N$.


$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 90$

Solution

By using formulas as well as treating $1^3+3^3+...+2019^3$ as $(1^3+2^3+...+2019^3)-(2^3+4^3+...+2018^3)$, we get the value is $(2019(1010))^2-8((1009)(505))^2+4(1009)(505)(673)$ which is congruent to $0-0+40$ which is answer choice $\boxed{(C)}$.

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