2014 AMC 10B Problem 17

Problem 17 of 25IntermediateAlgebra Number Theory

What is the greatest power of

2

that is a factor of

10^{1002} - 4^{501}?

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Solution

Factor out the obvious power of

2

:

10^{1002}-4^{501}=2^{1002}(5^{1002}-1)

. Since

5^{1002}-1=(5^{501}-1)(5^{501}+1)

, and

501

is odd,

5^{501}-1

is divisible by

4

but not by

8

, while

5^{501}+1

is divisible by

2

but not by

4

. Thus

5^{1002}-1

contributes exactly

2^3

, so the whole expression is divisible by

2^{1005}

but not

2^{1006}

. Thus, the correct answer is D .

More practice

Concepts: power of 2 · factoring · difference of squares

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.