2002 AMC 12P Problems/Problem 1

Problem

Which of the following numbers is a perfect square?

$\text{(A) }4^4 5^5 6^6 \qquad \text{(B) }4^4 5^6 6^5 \qquad \text{(C) }4^5 5^4 6^6 \qquad \text{(D) }4^6 5^4 6^5 \qquad \text{(E) }4^6 5^5 6^4$

Solution 1

For a positive integer to be a perfect square, all the primes in its prime factorization must have an even exponent. With a quick glance at the answer choices, we can eliminate options $\textbf{(A)}$ because $5^5$ is an odd power $\textbf{(B)}$ because $6^5 = 2^5 \cdot 3^5$ and $3^5$ is an odd power $\textbf{(D)}$ because $6^5 = 2^5 \cdot 3^5$ and $3^5$ is an odd power, and $\textbf{(E)}$ because $5^5$ is an odd power. This leaves option $\textbf{(C)},$ in which $4^5=2^2^5=2^10$ (Error compiling LaTeX. Unknown error_msg), and since 10, 4, and 6 are all even, it is a perfect square. Thus, our answer is $\box{\textbf{(C)} 4^4 5^4 6^6},$ (Error compiling LaTeX. Unknown error_msg)

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by First question	Followed by Problem 2
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
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

2002 AMC 12P Problems/Problem 1

Problem

Solution 1

See also