Square each integer $n$ in the range $1\le n\le 10$ and find the remainders when the squares are divided by $11$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $11$?
I want to find the remainders of the squares of the integers from $1$ to $10$ when divided by $11$. I can use the modulo notation to write this as $n^2 \mod 11$. I can use a calculator or do some mental arithmetic to find these remainders. For $n=1$, I get $n^2 \mod 11 = 1$. For $n=2$, I get $n^2 \mod 11 = 4$. For $n=3$, I get $n^2 \mod 11 = 9$. For $n=4$, I get $n^2 \mod 11 = 5$. For $n=5$, I get $n^2 \mod 11 = 3$. For $n=6$, I get $n^2 \mod 11 = 3$ again. For $n=7$, I get $n^2 \mod 11 = 5$ again. For $n=8$, I get $n^2 \mod 11 = 9$ again. For $n=9$, I get $n^2 \mod 11 = 4$ again. For $n=10$, I get $n^2 \mod 11 = 1$ again. So the distinct remainders are $1, 3, 4, 5, 9$. To find $m$, I need to add them up. I get $m = 1 + 3 + 4 + 5 + 9 = 22$. To find the quotient when $m$ is divided by $11$, I need to do long division or use a calculator. I get $m \div 11 = 2$ with a remainder of $0$. 