shopaholic wrote:
When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?
A. 12
B. 15
C. 20
D. 28
E. 35
Let’s list some integers that have a remainder of 3 when divided by 5:
3, 8, 13, 18, 23, …
Let’s list some integers that have a remainder of 4 when divided by 7:
4, 11, 18, 25, …
We see that 18 is common in both lists and is the smallest positive integer that satisfies both conditions. Thus, y could be 18. To get a value for x, we can simply add the LCM of 5 and 7, i.e, 35, to 18. So, x could be 53 (notice that 53 also satisfies both conditions). We see that in this case, x - y = 35, and of all the choices, only 35 is factor of x - y.
Alternate Solution:
Since x produces a remainder of 3 when divided by 5, we can write x = 5p + 3 for some integer p.
Since x produces a remainder of 4 when divided by 7, we can write x = 7q + 4 for some integer q.
Since y produces a remainder of 3 when divided by 5, we can write y = 5s + 3 for some integer s.
Since y produces a remainder of 4 when divided by 7, we can write y = 7r + 4 for some integer r.
If we subtract the third equality from the first, we obtain: x - y = 5p - 5s = 5(p - s). Thus, x - y is a multiple of 5.
If we subtract the fourth equality from the second, we obtain: x - y = 7q - 7r = 7(q - r). Thus, x - y is a multiple of 7.
Since x - y is a multiple of both 5 and 7, it is a multiple of the LCM of 5 and 7 as well, namely 35.
Answer: E
_________________
See why Target Test Prep is the top rated GMAT course on GMAT Club.
Read Our Reviews