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 all even powers of 5 more than two end in ... $625$ . Also, all odd powers of five more than 2 end in ... $125$ . Thus, $(5^{501} + 1)$ would end in ... $126$ 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 ... $124$ , contributing two powers of two to the final result.

Adding these extra $3$ powers 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
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2014 AMC 10B Problems/Problem 17

Problem 17

Solution

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