8. What is the largest prime factor of $106^{2}-15^{2}$ ?
A 3
B 7
C 11
D 13
E 17
SOLUTION
D

Note: It is not a good idea to attempt to calculate $106^{2}$ and $15^{2}$, then do a subtraction and finally attempt to factorize the resulting answer.

Instead we make use of the standard factorization of the difference of two squares:
$$
x^{2}-y^{2}=(x-y)(x+y) \text {. }
$$

We have
$$
\begin{aligned}
106^{2}-15^{2} & =(106-15)(106+15) \\
& =91 \times 121 \\
& =7 \times 13 \times 11 \times 11 .
\end{aligned}
$$

Therefore the prime factorization of $106^{2}-15^{2}$ is $7 \times 11^{2} \times 13$, from which we see that its largest prime factor is 13 .

For inVESTIGATION
8.1 What is the largest prime factor of $300^{2}-3^{2}$ ?