2014 AMC 10B Problems/Problem 17

Problem 17

What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$ ?

$\textbf{(A) } 2^{1002} \qquad\textbf{(B) } 2^{1003} \qquad\textbf{(C) } 2^{1004} \qquad\textbf{(D) } 2^{1005} \qquad\textbf{(E) }2^{1006}$

Solution

We begin by factoring the $2^{1002}$ out. This leaves us with $5^{1002} - 1$ . We factor the difference of squares, leaving us with $(5^{501} - 1)(5^{501} + 1)$ . We note that for all powers of 5 more than two, it ends in ... $25$ Thus, $(5^{501} + 1)$ would end in ... $26$ and thus would contribute one power of two to the answer, but not more. We can continue to factor $(5^{501} - 1)$ as a difference of cubes, leaving us with $(5^{167} - 1)$ times an odd number. $(5^{167} - 1)$ ends in ... $24$ , contributing two powers of two to the final result. Adding these extra 3 power of two to the original 1002 factored out, we obtain the final answer of $\textbf{(D) } 2^{1005}$ .

2014 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2014 AMC 10B Problems/Problem 17

Problem 17

Solution

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