Let w , x , y , and z be whole numbers. If 2^w · 3^x · 5^y · 7^z = 588 , then what does 2w + 3x + 5y + 7z equal?

AMC 8 2011 Problem 17

AMC8 2011 年第 17 题

整除与因数★★★☆☆

Let

w

x

y

, and

z

be whole numbers. If

2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588

, then what does 2w + 3x + 5y + 7z equal?

设

w

、

x

、

y

、

z

都是非负整数。若

2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588

，那么 2w + 3x + 5y + 7z 等于多少？

正确答案 · Correct Answer

Solution 1

The prime factorization of 588 is

2^2\cdot3\cdot7^2.

We can see w=2, x=1, and z=2. Because

5^0=1, y=0.

2w+3x+5y+7z=4+3+0+14=21

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