Bunuel wrote:

The product of integers w, x, y, and z is 210, and w > x > y > z > 0. Which of the following could NOT be the sum w + x + y + z?

A. 17

B. 19

C. 21

D. 31

E. 41

Let’s break down 210 as a product of two positive integers:

210 = 1 x 210 = 2 x 105 = 3 x 70 = 5 x 42 = 6 x 35 = 7 x 30 = 10 x 21 = 14 x 15

For each of the expressions except 1 x 210, we can write each factor as the product of two distinct factors. For example:

210 = 2 x 105 = (1 x 2) x (3 x 35) = (1 x 2) x (5 x 21) = (1 x 2) x (7 x 15)

From the above, we see that the sum w + x + y + z could be 41, 29, and 25, respectively. Thus, we can eliminate choice E.

Now let’s use 210 = 3 x 70:

210 = 3 x 70 = (1 x 3) x (2 x 35) = (1 x 3) x (5 x 14) = (1 x 3) x (7 x 10)

From the above, we see that the sum w + x + y + z could be 41, 23, and 21, respectively. Thus, we can eliminate choice C.

Now let’s use 210 = 5 x 42:

210 = 5 x 42 = (1 x 5) x (2 x 21) = (1 x 5) x (3 x 14) = (1 x 5) x (6 x 7)

From the above, we see that the sum w + x + y + z could be 29, 23, and 19, respectively. Thus, we can eliminate choice B.

Now let’s use 210 = 6 x 35:

210 = 6 x 35 = (1 x 6) x (5 x 7) = (2 x 3) x (1 x 35) = (2 x 3) x (5 x 7)

From the above, we see that the sum w + x + y + z could be 19, 41, and 17, respectively. Thus, we can eliminate choice A.

We can stop here since we’ve eliminated choices A, B, C, and E. Thus, D is the correct answer.

Answer: D

_________________

Jeffery Miller

Head of GMAT Instruction

GMAT Quant Self-Study Course

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