4. Which of the following is not a multiple of 5 ?
A $2019^{2}-2014^{2}$
B $2019^{2} \times 10^{2}$
C $2020^{2} \div 101^{2}$
D $2010^{2}-2005^{2}$
E $2015^{2} \div 5^{2}$
Solution
Using the factorization of the difference of two squares, we see that $2019^{2}-2014^{2}=$ $(2019-2014)(2019+2014)=5 \times 4033$, and hence is a multiple of 5 .
$2019^{2} \times 10^{2}$ is a multiple of $10^{2}$, that is, a multiple of 100 , and hence is a multiple of 5 . $2020^{2} \div 101^{2}=(20 \times 101)^{2} \div 101^{2}=20^{2}=400$ and so is a multiple of 5 .
Because $2010^{2}$ and $2005^{2}$ are both multiples of 5 , their difference, $2010^{2}-2005^{2}$, is also a multiple of 5 .
However, $2015^{2} \div 5^{2}=\left(\frac{2015}{5}\right)^{2}=403^{2}$ and, because 403 is not a multiple of $5,403^{2}$ is not a multiple of 5 .
Hence the correct answer is option E.