2001 CEMC Gauss (Grade 8) Problems/Problem 20

Problem

What number should be placed in the box to make $10^{4} \times 100^{\Box} = 1000^{6}$?

$\text{ (A) }\ 7 \qquad\text{ (B) }\ 5 \qquad\text{ (C) }\ 2 \qquad\text{ (D) }\ \text{3}{2} \qquad\text{ (E) }\ 10$

Solution 1

Let $x$ be the number represented by the box. To make the values easier to compute, we can make the exponents the same.

$10^{4} \times 100^{x} = 1000^{6}$

$10^{4} \times (10^{2})^{x} = (10^{3})^{6}$

$10^{4} \times 10^{2x} = 10^{18}$

$10^{2x + 4} = 10^{18}$

Equating the exponents, we have:

$2x + 4 = 18$

$2x = 14$

$x = \boxed {\textbf {(A) } 7}$

~anabel.disher

2001 CEMC Gauss (Grade 8) (ProblemsAnswer KeyResources)
Preceded by
Problem 19
Followed by
Problem 21
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CEMC Gauss (Grade 8)