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