1. What is $2015^{2}-2016 \times 2014$ ?
A -2015
B -1
C 0
D 1
E 2015
Solution
D
Commentary
This question could be answered by doing two multiplications, to evaluate $2015^{2}$ and $2016 \times 2014$, and then one subtraction. It should be clear that this method cannot be what was intended as it would be tedious and take up a lot of time. There must be a better method. The clue is that the product $2016 \times 2014$ may be rewritten as $(2015+1) \times(2015-1)$, and that this last expression is equivalent to the difference of two squares, namely, $2015^{2}-1^{2}$. This leads to the solution given below.
[See Question 22 for another case where the factorization of the difference of two squares is useful.]

We have
$$
\begin{aligned}
2015^{2}-2016 \times 2014 & =2015^{2}-(2015+1) \times(2015-1) \\
& =2015^{2}-\left(2015^{2}-1\right) \\
& =2015^{2}-2015^{2}+1 \\
& =1 .
\end{aligned}
$$