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integer question

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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integer question

by sidon » Sat Apr 10, 2010 10:11 am
Hi,

Can anyone help solving this integer question ?


If k is an integer, and (35^2-1)/k is an integer, then k could be each of the following, EXCEPT
(A) 8
(B) 9
(C) 12
(D) 16
(E) 17



Thanks,
Sidon
Last edited by sidon on Sat Apr 10, 2010 10:26 am, edited 1 time in total.
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by 4GMAT_Mumbai » Sat Apr 10, 2010 10:14 am
Hi,

Did you by any chance miss 'k' in the expression '(35^2-1)/2' ...
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by sidon » Sat Apr 10, 2010 10:28 am
oops , sorry , you are right .

the question is (35^2-1)/k .

I have edited the first post ...

Thanks.
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by 4GMAT_Mumbai » Sun Apr 11, 2010 6:38 pm
Let x = (35^2-1)/k
x = (35 + 1) * (35 -1) / k = 36 * 34 / k

B cannot be the answer since 36 has a factor of 9 in it. x will be an integer
C cannot be the answer since 36 has a factor of 12 in it. x will be an integer
E cannot be the answer since 34 has a factor of 17 in it. x will be an integer
A cannot be the answer since 36 * 34 = 4 * 9 * 2 * 17 = (4 * 2) * 9 * 17 = 8 * 9 * 17. x will be an integer.

D is the answer because the numerator does not have a factor of 16.

Hope this helps.
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by sidon » Mon Apr 12, 2010 10:22 am
thanks , good answer ...

sidon
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by Rahul@gurome » Wed Apr 14, 2010 9:50 am
4GMAT_Mumbai's answer is a good one. I would also like to suggest a slightly different method of solving this.

To simplify: (35^2-1)/k

========
Step 1) There is a shortcut to figure out the square of an integer ending with '5'.
Drop the ending '5', and multiply the (remaining number) with (remaining number + 1)
Then append '25' at the end of the result.

So:
35^2
= 3 * (3+1) <== Drop the ending '5', and multiply the (remaining number) with (remaining number + 1)
= 3*4
= 12
= 1225 <== Then append '25' at the end of the result.

Step 2)
(35^2-1)/k can be then written as:
(1225 - 1)/k
= 1224/k

Step 3)
Factorize 1224 into all it's prime factors

1224
2 * 612
2 * 2 * 306
2 * 2 * 2 * 153
2 * 2 * 2 * 3 * 51
2 * 2 * 2 * 3 * 3 * 17

Step 4)
(2 * 2 * 2 * 3 * 3 * 17) / k

Step 5) Now you can plug in the answer choices for 'k' one by one into the equation above and check which one DOES NOT result in an integer.

(A) 8 <-- 8 =2*2*2 which goes evenly in the numerator
(B) 9 <-- 9 = 3*3 which goes evenly in the numerator
(C) 12 <-- 3*2*2 which goes evenly in the numerator
(D) 16 <-- 16 = 2*2*2*2 - DOES NOT GO EVENLY IN THE NUMERATOR - This is our answer.
(E) 17 <-- goes evenly in the numerator
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by shahdevine » Tue Apr 20, 2010 12:11 pm
sidon wrote:Hi,

Can anyone help solving this integer question ?


If k is an integer, and (35^2-1)/k is an integer, then k could be each of the following, EXCEPT
(A) 8
(B) 9
(C) 12
(D) 16
(E) 17



Thanks,
Sidon
I would look at (35^2-1) as difference of squares and simplify --> a^2-b^2=(a+b)(a-b) --> (35-1)(35+1)=34*36

then back solve to see if any of the options for k go into 34*36 evenly. After exhausting all the answers very quickly you will see 16 does not evenly go into 34*36. D is answer.

You got this!
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