13. Anish has a number of small congruent square tiles to use in a mosaic. When he forms the tiles into a square of side $n$, he has 64 tiles left over. When he tries to form the tiles into a square of side $n+1$, he has 25 too few.
How many tiles does Anish have?
A 89
B 1935
C 1980
D 2000
E 2019
Solution
D
Because Anish has 64 tiles left over when he forms a square of side $n$, he has $n^{2}+64$ tiles.
Because Anish has 25 tiles too few to make a square of side $n+1$, he has $(n+1)^{2}-25$ tiles.
Therefore $n^{2}+64=(n+1)^{2}-25$. Now
$$
\begin{aligned}
n^{2}+64=(n+1)^{2}-25 & \Leftrightarrow n^{2}+64=\left(n^{2}+2 n+1\right)-25 \\
& \Leftrightarrow 2 n=64+25-1 \\
& \Leftrightarrow 2 n=88 \\
& \Leftrightarrow n=44 .
\end{aligned}
$$

Because $n=44$, the number of tiles that Anish has is given by $44^{2}+64=1936+64=2000$.