2016 AMC 8

Complete problem set with solutions and individual problem pages

Problem 15 Medium

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

  • A.

    8

  • B.

    16

  • C.

    32

  • D.

    64

  • E.

    128

Answer:C

First, we use difference of squares on 13^4 - 11^4 = (13^2)^2 - (11^2)^2 to get 13^4 - 11^4 = (13^2 + 11^2)(13^2 - 11^2). Using difference of squares again and simplifying, we get (169 + 121)(13+11)(13-11) = 290 \cdot 24 \cdot 2 = (2\cdot 8 \cdot 2) \cdot (3 \cdot 145). Realizing that we don't need the right-hand side because it doesn't contain any factor of 2, we see that the greatest power of 2 that is a divisor 13^4 - 11^4 is \boxed{\textbf{(C)}\ 32}.

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Problem 13 2016 AMC 8
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