x and y are positive integers. When 16x is divided by y,

Question

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

EDIT: Changed answer choice E

BrentGMATPrepNow · Accepted Answer

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

RELATED VIDEO FROM OUR COURSE
 https://www.youtube.com/watch?v=XCxgIR042tE

rohit8865 · Answer

16x = yx+4
y=(16x-4)/x

if x=1 then y =12
if x= 2 then y= 14
if x=3 y is no longer integer
if x=4 then y = 15

upto here y = 12+14+15 = 41...
but i m not finding any such option.....

can any one explain!!!!!

thanks

BrentGMATPrepNow · Answer

Yeesh, I added those values several times and got 39 each time. 
I've changed answer choice E to the correct value.

Cheers and thanks, 
Brent

vitaliyGMAT · Answer

Hi It seems I've missed something. 16x = yx + 4 16x - yx = 4 x(16 - y) = 4 (1*4), (2*2) and (4*1). I'm discarding negativ values because our x an y are >0. x=1, 16 - y = 4 ---> y=12, check: 16*1 = 12*1 + 4 x = 2, 16 - y = 2 -----> y = 14, check: 32 = 14*2 + 4 x = 4, 16 - y = 1 -------> y=15, chek: 64 = 15*4 + 4 4

BrentGMATPrepNow · Answer

You're correct, vitaliyGMAT It seems I have not yet mastered addition. :oops:

Cheers, Brent

alandizzle · Answer

Great question! I solved it a very similar way to you all.
(16x)/y = x + 4
--> 16x = xy + 4 --> y = (16x - 4)/x ---> y = 16 - 4/x.

We know from the prompt that Y is an integer, therefore, the only values of X that will make Y and integer is 0, 1, 2, 4. However, the prompt says that X & Y are  positive integers . Therefore we can't use 0. So plugging in the values of 1, 2, and 4 into the equation, you're left with 15 + 14 + 12 = 41.

BrushMyQuant · Answer

When 16x is divided by y, the quotient is x, and the remainder is 4

When a positive integer m is divided by 17, the remainder is 3

Theory: Dividend = Divisor*Quotient + Remainder

16x -> Dividend
y -> Divisor
x -> Quotient
4 -> Remainders
=> 16x = y*x + 4 = xy + 4

=> y =  \frac{16x - 4}{x}  =  \frac{16x}{x} - \frac{4}{x}  =  16 - \frac{4}{x}

Since y is an integer =>  \frac{4}{x}  will be an integer
=> 4 should be divisible by x
=> x = 1, 2 or 4

x = 1  
16x = xy + 4
=> 16*1 = 1*y + 4
=> y = 16 - 4 = 12

x = 2  
16x = xy + 4
=> 16*2 = 2*y + 4
=> y =  \frac{32 - 4}{2} = 14

x = 4  
16x = xy + 4
=> 16*4 = 4*y + 4
=> y =  \frac{64 - 4}{4} = 15

=> Sum of all values of y = 12 + 14 + 15 = 41

So,  Answer will be E 
Hope it helps!

Watch the following video to learn the Basics of Remainders

https://www.youtube.com/watch?v=P0S6j2uTaj4

rxb266 · Answer

We can write the equation and solve -

y(x) + 4 = 16(x)

or y = 16 + (4/x)

since y is an integer, 4/x has to be integer. Hence possible values of x are: 1, 2 and 4. Therefore the possible values of y are - 12, 14 and 15. Therefore the sum of all the possible values of y is : 41  (Option E)

Kinshook · Answer

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

16x/y = x + 4/y
16x = xy + 4
y = (16x-4)/x = 16 - 4/x
x=1; y=12
x=2; y=14
x=4; y=15

Sum of all possible y-values = 12 + 14 + 15 = 41

IMO E

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x and y are positive integers. When 16x is divided by y,

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