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The value of a+b: from Princetonreview

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The value of a+b: from Princetonreview

by torofish » Thu Dec 03, 2009 2:13 am
The answer is B

My answer is E.

I get that A is not the answer.

But for 2
(a-b)(a+b) = -25

I thought that
It can be -5 and +5 to get -25.
So we couldn't know the answer.

But in solution it says
(a+b)(a-b) = (25)(-1)
with the restriction a and b is positive
it's 25.

.....does "a and b are positive integers" play an important role?
please explain, I don't really get its solution.
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by BuckeyeT » Thu Dec 03, 2009 6:24 am
Can you post the complete question? I don't have access to Princetonreview.

But when a statement tells you that "a and b are positive integers", it is ALWAYS important. In this case, you will know...

a>0
b>0
(a+b) > (a-b)

If (a-b)(a+b) = -25, we know that one of the values (a-b) or (a+b) must be negative. And since we're dealing with integers, we have a limited number of factors.
1 * -25
-1 * 25
5 * -5
-5 * 5

Remember above that (a+b) > (a-b) since we're dealing with POSITIVE integers.
So, a+b must be the positive value. And, a-b must be the negative value.

a+b = 1
a-b = -25
a=1-b --> 1-b-b = -25 --> -2b = -26 --> b=13
a+13 = 1 --> a= -12 This negates our statement that both are POSITIVE integers.

a+b = 25
a-b = -1
a= 25-b --> 25-b-b = -1 --> -2b = -26 --> b=13.
a-13 = -1 --> a=12. This works as both a and b are POSITIVE integers.

a+b = 5
a-b = -5
a = 5-b --> 5-b-b = -5 --> -2b = 0 --> b=0. We don't have to prove further because b cannot be 0. It must be positive!

So since we only have one solution that works, we can say that (2) is SUFFICIENT.

Do you see how important knowing they are POSITIVE is?
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by hariharakarthi » Thu Dec 03, 2009 6:48 am
statment 1:

a^2 - 2ab + b^2 = 25
(a-b)^2=25

Write a-b = x, then the equation becomes x^2 = 25
|x|=5.

so, a-b = 5 or a-b = -5
From this we can not find out value of a+b. Hence, INSUFFICIENT.

statement 2:
a^2 - b^2 = -25
(a+b)(a-b) = -25
a+b = -25/ (a-b)
Hence, INSUFFICINET.

Now, combine both 1 and 2,
1 -> a-b= 5 or a-b = -5
2 -> a+b = -25/(a-b)
We know that a and b are positive integers, a-b should be negative,

sub 1 in 2, we get a+b =-25/-5

Note: while substituting values for a-b we need to be careful.

So, ANS is C

Regards,
hhk
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by BuckeyeT » Thu Dec 03, 2009 6:55 am
hariharakarthi-

See my answer above as to why B is sufficient by itself. Since a and b are positive integers, we can determine the possible factors of -25 as shown. From there, we can prove that only 1 set of values work.
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by mp2437 » Thu Dec 03, 2009 6:57 am
The reason they say (a-b)(a+b) is (-1)(25) is because that is the only possibility you get when a and b are positive integers. To get -5 and +5, then a = 0 and b = -5, but a cant be 0 since they tell you it has to be a positive integer.
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by hariharakarthi » Thu Dec 03, 2009 6:59 am
Oops..Thanks BuckeyeT. I did not see your explanation. I just solved and posted my comments. Thank you very much.

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by torofish » Thu Dec 03, 2009 11:16 pm
Thank you. I kinda get that now.

The whole question is at the bottom of the question. I posted as a picture.
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