2004 AMC 12A Problem 6

Problem 6 of 25EasierAlgebra

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?

Show solution

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.