Bunuel wrote:
How many pairs positive integers x and y satisfy 2x + y < 40, if y is a multiple of x?
A. 90
B. 91
C. 92
D. 93
E. 94
Since x and y are positive integers and y is a multiple of x, then y ≥ x.
If x = 1, we have:
2(1) + y < 40
y < 38
So y can be any integer from 1 to 37, inclusive. Therefore, there are 37 pairs of positive integers x and y (or solutions) when x = 1.
If x = 2, we have:
2(2) + y < 40
y < 36
So y can be any even integer from 2 to 34, inclusive. Therefore, there are 17 solutions when x = 2.
If x = 3, we have:
2(3) + y < 40
y < 34
So y can be any multiple of 3 from 3 to 33, inclusive. Therefore, there are 11 solutions when x = 3.
If x = 4, we have:
2(4) + y < 40
y < 32
So y can be any multiples of 4 from 4 to 28, inclusive. Therefore, there are 7 solutions when x = 4.
If x = 5, we have:
2(5) + y < 40
y < 30
So y can be any multiples of 5 from 5 to 25, inclusive. Therefore, there are 5 solutions when x = 5.
If x = 6, we have:
2(6) + y < 40
y < 28
So y can be any multiples of 6 from 6 to 24, inclusive. Therefore, there are 4 solutions when x = 6.
If x = 7, we have:
2(7) + y < 40
y < 26
So y can be any multiples of 7 from 7 to 21, inclusive. Therefore, there are 3 solutions when x = 7.
If x = 8, we have:
2(8) + y < 40
y < 24
So y can be any multiples of 8 from 8 to 16, inclusive. Therefore, there are 2 solutions when x = 8.
If x = 9, we have:
2(9) + y < 40
y < 22
So y can be any multiples of 9 from 9 to 18, inclusive. Therefore, there are 2 solutions when x = 9.
If x = 10, we have:
2(10) + y < 40
y < 20
So y can only be 10. Therefore, there is only 1 solution when x = 10.
Similarly, when x = 11, 12 or 13, there is 1 solution for each of these x-values. If x ≥ 14, then there are no solutions. (For example, if x = 14, 2(14) + y < 40 → y < 12. However, recall that y ≥ x since y is a multiple of x.)
Therefore, the total number of solutions is:
37 + 17 + 11 + 7 + 5 + 4 + 3 + 2 x 2 + 1 x 4 = 54 + 18 + 12 + 4 + 4 = 72 + 20 = 92
Answer: C
_________________
★
★
★
★
★
214 REVIEWS
5-STAR RATED ONLINE GMAT QUANT SELF STUDY COURSE
NOW WITH GMAT VERBAL (BETA)
See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews