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

Attempt Problem 6 of the 2004 AMC 12A below, then check your answer against the professionally curated solution 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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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

Factor each difference to write it as a multiple of a power of

2004

U - V = 2004^{2005},

while every other choice is a multiple of the smaller

2004^{2004}

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.