Let U = 2 × 2004^2005, V = 2004^2005, W = 2003 × 2004^2004, X = 2 × 2004^2004, Y = 2004^2004 and Z = 2004^2003. Which of the following is largest?

2004 AMC 12A Problem 6 — Solution

2004 AMC 12A Problem 6

Below is the professionally curated solution for Problem 6 of the 2004 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12A solutions, or check the answer key.

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Concepts:factoring exponent

Difficulty rating: 1220

Let $U = 2 \cdot 2004^{2005},$ $V = 2004^{2005},$ $W = 2003 \cdot 2004^{2004},$ $X = 2 \cdot 2004^{2004},$ $Y = 2004^{2004}$ and $Z = 2004^{2003}.$ Which of the following is largest?

$U - V$

$V - W$

$W - X$

$X - Y$

$Y - Z$

Solution:

Compute each difference by factoring:

$U - V = 2004^{2005},$ $V - W = 2004^{2004},$ $W - X = 2001 \cdot 2004^{2004},$ $X - Y = 2004^{2004},$ and $Y - Z = 2003 \cdot 2004^{2003}.$

Since $2004^{2005} = 2004 \cdot 2004^{2004}$ exceeds each of the others, none of which reaches $2004^{2005},$ the difference $U - V$ is the largest.

Thus, the correct answer is A.

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