2016 AMC 8 Problem 15

Attempt Problem 15 of the 2016 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

15.

What is the largest power of $2$ that is a divisor of $13^4 - 11^4?$

$8$

$16$

$32$

$64$

$128$

Click to load, then click again to play

We can factor this expression using difference of squares.

$\begin{aligned} 13^4-11^4 &=(13^2+11^2)\\ &\qquad\cdot(13^2-11^2)\\ &=290\cdot48\\ &=32\cdot435. \end{aligned}$

Because $435$ is odd, $32$ is the largest power of $2$ that divides the expression.

Thus, C is the correct answer.