4. Which of the following is equal to $2017-\frac{1}{2017}$ ?
A $\frac{2017^{2}}{2016}$
B $\frac{2016}{2017}$
C $\frac{2018}{2017}$
D $\frac{4059}{2017}$
E $\frac{2018 \times 2016}{2017}$
Solution
E

Writing both 2017 and $\frac{1}{2017}$ over a common denominator, we have
$$
2017-\frac{1}{2017}=\frac{2017^{2}-1}{2017} \text {. }
$$

Now,
$$
2017^{2}-1=2017^{2}-1^{2} .
$$

Hence, using the standard factorization of the difference of two squares, we have
$$
2017-\frac{1}{2017}=\frac{2017^{2}-1^{2}}{2017}=\frac{(2017+1)(2017-1)}{2017}=\frac{2018 \times 2016}{2017} .
$$