Find the number of ordered integral solution of the following equation

Question

\frac{1}{x}+\frac{1}{y} = \frac{5}{11} , where x,y≠0

A. 0
B. 1
C. 2
D. 3
E. infinite

Kinshook · Answer

Asked: Find the number of ordered integral solution of the following equation

\frac{1}{x}+\frac{1}{y} = \frac{5}{11} , where x,y≠0

1/x = 5/11 - 1/y = (5y-11)/11y
x= 11y/(5y-11)

y = 2; x = 22/(-1) = -22; 1 Solution
y = -22; x = 2; 1 solution
y = 3; x = 33/4
y = 4; x = 44/9
y = 5; x = 55/14
y = 6; x = 66/19
y = 7; x = 77/24
y = 8; x = 88/29
y = 9; x = 99/34 
y = 10; x = 110/39
y = 11; x = 121/44 = 11/4

{x,y} = {(2,-22),(-22,2)}

IMO C

boomtangboy · Answer

Hi, Could you elaborate on what is an integral solution. The term is quite new to me and I wanted to know more. Also is this a official GMAT question ?

shameekv1989 · Answer

I feel it is B

For x = 2, y=-22 you can have a solution.

Kinshook · Answer

Hi  shameekv1989 
Since x & y are interchangeable
You will have 2 solutions

IMO C

shameekv1989 · Answer

Kinshook 
What you mean is since its "ordered integral solutions" - we can have (x,y) as 2,-22 and -22,2 as separate solutions?

I see some questions mention "how many pairs of integral solutions" - what do we take there? Specifically, if you could help me identifying when do I take pairs as different and when do I take them as same?

sujoykrdatta · Answer

Let us first understand what an "ordered integral solution" means and what "integral pairs of solutions" means: Say: a + b = 6, where a and b are positive integers The ordered solutions are : a=5,b=1; a=4,b=2; a=3,b=3; a=2,b=4; a=1,b=5 => There are 5 ordered solutions Number of pairs of solutions : 5,1; 4,2; 3,3 => There are 3 pairs of solutions I hope the significance of the word "ordered" is clear Now, let us solve the given question. Please note this is a complicated equation to solve since there is no restriction on either x or y. Hence, if we check with values of y (and try to find when x becomes an integer), we would have to keep on doing it. For example, we cannot say for certain, that for y = 55, we will not get a solution or for y = -33, we won't get some solution. Thus, I have done (tried to do

) an analytical approach - I firmly believe that this is beyond the scope of the GMAT. This question could be relevant to the GMAT but with restrictions on the values of x and y so that we can stop after a certain number of iterations. 1/x + 1/y = 5/11 => 1/x = 5/11 - 1/y = (5y - 11)/11y => x = 11y/(5y - 11) Since x and y are integers (positive or negative), we must ensure that y is divisible by (5y - 11) or 11 is divisible by (5y - 11): Case 1: 11 divided by (5y - 11) gives an integer value: => 5y - 11 = 11 or -11 => y = 0 (not possible - cannot divide by 0) or y = 4.4 (non-integer - ignored) => We have no solutions Case 2: y divided by (5y - 11) gives an integer => only possible if |y| > |5y - 11| => 11/6 < y < 3 i.e. y = 2: # y = 2: 2 divided by -1 gives an integer (correct) => y = 2, x = -22 (=> The other pair is: y = -22 and x = 2) => We have 2 solutions Case 3: There might be cases where (5y - 11) has factors which partly cancel with 11 and partly cancel with y resulting in an integer value This would imply that 5y - 11 and y have some GCD (not equal to 1) GCD of y and 5y - 11 = GCD of 5y and 5y - 11 = GCD of 5y and 5y - (5y - 11) ... (since GCD of A and B is the same as GCD of A and A-B) = GCD of 5y and 11 The above GCD (greater than 1) will exist only if y = a multiple of 11 (say 11k) => x = 11y/(5y - 11) = 121k/(55k - 11) = 11k/(5k - 1) The only solution for this is k = -2 (i.e. y = -22: we have already obtained that above) Thus, there are 2 solutions Answer C

nick1816 · Answer

1/x + 1/y = 5/11

y = \frac{11x}{(5x - 11)}

/(5x-11) = y

y =  \frac{1}{5}

Since y is an integer, 5x-11 must be a factor of 121 and     is a multiple of 5.

5x-11 can be -121, -11, -1, 1, 11 or 121.

If 5x-11 is equal to 1, 11 or 121, x will not have integral value . So we can discard these cases.

Case 1- 
5x-11= -121
x=-22, y =2

Case 2- 
5x-11= -11
x=0; y=0 (Not possible)

Case 3- 
5x-11=-1
x=2; y=-22

2 solutions are possible

chetan2u · Answer

\frac{1}{x}+\frac{1}{y} = \frac{5}{11}..........\frac{x+y}{xy}=\frac{5}{11}.......11(x+y)=5xy.......11x=(5x-11)y.......y=\frac{11x}{5x-11}

Now 5x-11 should be a factor of 11x... Will be possible only when 
 5x-11 = 1 , but x will be a fraction
 5x-11=-1...x=2  and  y=\frac{11*2}{-1}=-22 
So two ordered integral solutions from it (x,y)=(2,-22) and (-22,2)
rest nothing possible

However not something that would be in GMAT as of date.

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Find the number of ordered integral solution of the following equation

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