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# If x and y are positive integers and 1 + x + y +xy = 21

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If x and y are positive integers and 1 + x + y +xy = 21 [#permalink]  11 Jun 2012, 07:20
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If x and y are positive integers and 1 + x + y +xy = 21, what is the value of x?

(1) y>3
(2) y=6

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Re: If x and y are positive integers and 1 + x + y +xy = 21 [#permalink]  11 Jun 2012, 07:40
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Good question. +1.

If x and y are positive integers and 1 + x + y +xy = 21, what is the value of x?

$$1 + x + y +xy = 21$$ --> $$(x+1)(y+1)=21$$ --> either $$x+1=3$$ and $$y+1=7$$ OR $$x+1=7$$ and $$y+1=3$$. Notice that $$x+1=1$$ and $$y+1=21$$ OR $$x+1=21$$ and $$y+1=1$$ is not possible since in this case either $$x$$ or $$y$$ equals zero and we are told that $$x$$ and $$y$$ are positive integers.

Now, from $$x+1=3$$ and $$y+1=7$$ --> $$x=2$$ and $$y=6$$ AND from $$x+1=7$$ and $$y+1=3$$ --> $$x=6$$ and $$y=2$$.

(1) y>3 --> $$y=6$$, so $$x=2$$. Sufficient.
(2) y=6 --> $$x=2$$. Sufficient.

Hope it's clear.
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Re: If x and y are positive integers and 1 + x + y +xy = 21 [#permalink]  11 Jun 2012, 07:29
Hi,

We have, 1 + x + y +xy = 21 (where x & y are natural numbers/positive integers)
or x(1+y) = 20 -y
or x = (20-y)/(1+y)

Using (1)
y > 3, i.e., y = 4, 5, 6...
when y=4, x = (20-4)/5 = 16/5 (not a natural number)
when y=5, x = (20-5)/6 = 15/6 (not a natural number)
when y=6, x = (20-6)/7 = 14/7 = 2
when y=7, x = (20-7)/8 = 13/8 (not a natural number), and for remaining values of y, there would be no positive integral value of x.
so, (x,y) = (2,6). Sufficient

Using (2)
y = 6, => x = (20-6)/(1+6) = 2
so, (x,y) = (2,6). Sufficient.

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Re: If x and y are positive integers and 1 + x + y +xy = 21 [#permalink]  13 Mar 2015, 12:27
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Re: If x and y are positive integers and 1 + x + y +xy = 21   [#permalink] 13 Mar 2015, 12:27
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