Data Sufficiency

If x and y are positive integers, what is the value of x?

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5
2) When 5y is divided by x, the quotient is y, and the remainder is 0

Answer: D

Source: www.gmatprepnow.com
Difficulty level: 700+

Brent@GMATPrepNow wrote:If x and y are positive integers, what is the value of x?

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5
2) When 5y is divided by x, the quotient is y, and the remainder is 0

Target question: What is the value of x?

Statement 1: When 9x is divided by 2y, the quotient is x, and the remainder is 5
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

So, from the above rule, we can write: 9x = 2yx + 5
Rewrite this as: 9x - 2yx = 5
Factor out the x to get: x(9 - 2y) = 5
Since x and 9 - 2y MUST BE INTEGERS, and since their PRODUCT is 5, we know that EITHER x = 1 and 9 - 2y = 5 OR x = 5 and 9 - 2y = 1

If x = 1 and 9 - 2y = 5, then x = 1 and y = 2
If x = 5 and 9 - 2y = 1, then x = 5 and y = 4

At this point it LOOKS LIKE there are two possible values for x. However, when we test these values, we have a problem.

If we plug x = 1 and y = 2 into statement 1, we get: When 9(1) is divided by 2(2), the quotient is 1, and the remainder is 5. We can see this this is NOT TRUE. So, x = 1 and y = 2 is NOT A SOLUTION

If we plug x = 5 and y = 4 into statement 1, we get: When 9(5) is divided by 2(4), the quotient is 5, and the remainder is 5. In other words, When 45 is divided by 8, the quotient is 5, and the remainder is 5. This WORKS. So, x = 5 and y = 4 IS A SOLUTION

So, we can conclude that x = 5
Since we can answer the target question with certainty, statement 1 is SUFFICIENT


Statement 2: When 5y is divided by x, the quotient is y, and the remainder is 0
So, from the above rule, we can write: 5y = xy + 0
Rewrite this as: 5y - xy = 0
Factor: y(5 - x) = 0
So, either y = 0, or x = 5
Since we're told y is POSITIVE, we know that y does not equal 0, which means x must equal 5
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent