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# If x and y are positive integers such that x^2 - y^2 = 23. What is the

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Intern
Joined: 31 Jan 2018
Posts: 9
If x and y are positive integers such that x^2 - y^2 = 23. What is the  [#permalink]

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03 Feb 2018, 22:18
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If x and y are positive integers such that x^2 - y^2 = 23. What is the value of y?

A. 10
B. 11
C. 12
D. 23
E. Cannot be determined
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Joined: 02 Aug 2009
Posts: 6519
Re: If x and y are positive integers such that x^2 - y^2 = 23. What is the  [#permalink]

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03 Feb 2018, 22:59
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ggpapas wrote:
If x and y are positive integers such that $$x^2$$ - $$y^2$$ = 23

What is the value of y?

A.10
B.11
C.12
D.23
E. Cannot be determined

$$x^2-y^2=23....(x-y)(x+y)=23$$
But 23 is prime, so 23=1*23..
Since both x and y are POSITIVE..
x+y=23
x-y=1....
Solve this x=12 and y=11
B
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Re: If x and y are positive integers such that x^2 - y^2 = 23. What is the  [#permalink]

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03 Feb 2018, 23:13
ggpapas wrote:
If x and y are positive integers such that $$x^2$$ - $$y^2$$ = 23

What is the value of y?

A.10
B.11
C.12
D.23
E. Cannot be determined

$$x^2$$ - $$y^2$$ = $$23$$

Or, $$(x + y)(x - y) = 23$$

Or, $$(12 + 11)(12 - 11) = 23$$

So,Answer Y must be (B) 11
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If x and y are positive integers such that x^2 - y^2 = 23. What is the  [#permalink]

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04 Feb 2018, 14:40
ggpapas wrote:
If x and y are positive integers such that x^2 - y^2 = 23. What is the value of y?

A. 10
B. 11
C. 12
D. 23
E. Cannot be determined

Although using the difference of squares identity is probably more efficient, you can also use answer choices.

Simply square the integers given.*

A) 10. $$10^2 = 100$$
B) 11. $$11^2 = 121$$ (121 - 100) = 21
C) 12. $$12^2 = 144$$ (144 - 121) = 23
That's a match.

The numbers must be
x = 12, y = 11, because
$$(144 - 121) = (12^2 - 11^2) = 23$$

y = 11

*23 is positive. LHS is (positive) - (positive).
x must be greater than y. Hence there is no need to worry about squaring any integer less than 10 (Answer A).

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Re: If x and y are positive integers such that x^2 - y^2 = 23. What is the  [#permalink]

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05 Feb 2018, 19:43
Another approach is to start with option B (Manhattan Guide talks about this in more detail) and back solve

X^2-11^2 = 23
X^2-121 = 23
X^2 = 144
X= 12

Bingo! Answer becomes straight B!
Re: If x and y are positive integers such that x^2 - y^2 = 23. What is the &nbs [#permalink] 05 Feb 2018, 19:43
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# If x and y are positive integers such that x^2 - y^2 = 23. What is the

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