1. What is the value of $123^{2}-23^{2}$ ?
A 10000
B 10409
C 12323
D 14600
E 15658
Solution
D
Commentary
The SMC is a no calculator paper. This should be a clue that there is a better way to answer this question than separately squaring 123 and 23 and then doing a subtraction.
There is no great virtue in being able to evaluate $123^{2}-23^{2}$ without a calculator, but the underlying method of factorizing the difference of two squares $x^{2}-y^{2}$ as $(x-y)(x+y)$ is very useful and should be remembered.

We evaluate $123^{2}-23^{2}$ as follows.
$$
\begin{aligned}
123^{2}-23^{2} & =(123-23)(123+23) \\
& =100 \times 146 \\
& =14600
\end{aligned}
$$