Tricky: When 16x is divided by y, the quotient

[Brent@GMATPrepNow]

x and y are positive integers. When 16x is divided by y, the quotient is x, and the remainder is 4. What is the sum of all possible y-values?
A) 7
B) 12
C) 19
D) 26
E) 41

Answer: E

Source: GMAT Prep Now
Difficulty level: 700+

[Brent@GMATPrepNow]

x and y are positive integers. When 16x is divided by y, the quotient is x, and the remainder is 4. What is the sum of all possible y-values?

A) 7
B) 12
C) 19
D) 26
E) 41

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

------NOW ONTO THE QUESTION------------------------------

When 16x is divided by y, the quotient is x, and the remainder is 4.
Applying the above rule, we can write: 16x = (y)(x) + 4
Subtract xy from both sides: 16x - xy = 4
Factor: x(16 - y) = 4

Since x and (16 - y) are both positive integers, there are 3 possible solutions:
#1: x = 1 and (16 - y) = 4, in which case y = 12
#2: x = 2 and (16 - y) = 2, in which case y = 14
#3: x = 4 and (16 - y) = 1, in which case y = 15

What is the sum of all possible y-values?
SUM = 12 + 14 + 15
= 41

Answer: E

[Matt@VeritasPrep]

Brent, you are killing it with these great questions! I'm starting to worry that the GMAC is going to swoop in to hire you any day now and steal you away from us.

I've got more than one approach. Let's start with one that's friendly enough: plugging in numbers to figure out what's going on.

To find a solution, let's start with the first positive integer, x = 1. That gives us 16 / y = 1 with remainder 4, so y = 12. OK, great start!

Now let's try x = 2. That gives us 32 / y = 2, with remainder 4. Solving, we find y = 14. Still going strong!

Now let's try x = 3. That gives us 48 / y = 3, with remainder 4. Solving, we find no integer solution. Hmm.

Now let's try x = 4. That gives us 64 / y = 4, with remainder 4. Solving, we find y = 15.

At this point we've got 12 + 14 + 15 = 41, and there are no higher answers, so we're done.

Pretty mindless, but it's quick, effective, and gets you the answer to a problem that might have a high difficulty rating on test day.

[Matt@VeritasPrep]

Now let's go for a conceptual approach that I think is pretty spiffy.

We know that Dividend = Quotient*Divisor + Remainder, so let's start there:

16x = xy + 4

Now let's isolate y, since it's the variable we're asked for:

(16x - 4)/x = y

16 - (4/x) = y

Since y must be an integer, x must be a factor of 4. There are only three possibilities: x = 1, x = 2, and x = 4. Plugging those in, we get y = 16 - (4/1), y = 16 - (4/2), and y = 16 - (4/4), or y = 12, y = 14, and y = 15.

[Brent@GMATPrepNow]

I'm starting to worry that the GMAC is going to swoop in to hire you any day now and steal you away from us.

Ha! That's the dream, Matt!

[Jeff@TargetTestPrep]

x and y are positive integers. When 16x is divided by y, the quotient is x, and the remainder is 4. What is the sum of all possible y-values?
A) 7
B) 12
C) 19
D) 26
E) 41

We can create the following equation:

16x/y = x + 4/y

Multiplying the equation by y, we have:

16x = xy + 4

16x - xy = 4

x(16 - y) = 4

16 - y = 4/x

We can see that x is a factor of 4 (i.e., 4/x must be an integer), so x can be 1, 2, or 4.

If x = 1, then 16 - y = 4, i.e., y = 12.
If x = 2, then 16 - y = 2, i.e., y = 14.
If x = 4, then 16 - y = 1, i.e., y = 15.

Thus the the sum of all possible y-values is 12 + 14 + 15 = 41.

Answer: E