For some integer q, q^2 - 5 is divisible by all of the following EXCEPT
(A) 29
(B) 30
(C) 31
(D) 38
(E) 41
(A) 29
(B) 30
(C) 31
(D) 38
(E) 41
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Source: Beat The GMAT — Problem Solving |
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theCodeToGMAT
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is the Answer [spoiler]{B}[/spoiler]?
Last edited by theCodeToGMAT on Wed Oct 09, 2013 5:22 am, edited 2 times in total.
R A H U L
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Brent@GMATPrepNow
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There's a nice rule that goes something like this:[email protected] wrote:For some integer q, q² - 5 is divisible by all of the following EXCEPT
(A) 29
(B) 30
(C) 31
(D) 38
(E) 41
If A is divisible by k, but B is not divisible by k, then A+B is NOT divisible by k, and A-B is NOT divisible by k.
There's another rule that says, "If there are n consecutive integers, then 1 of them is divisible by n"
Okay, now onto the question.
IMPORTANT: Notice that only answer choice B is divisible by 3. We'll use this fact in our solution.
Rewrite the given expression: q² - 5 = q² - 1 - 4 = (q - 1)(q + 1) - 4
NOTICE that (q - 1), q and (q + 1) are 3 consecutive integers, so one of them must be divisible by 3 (from the red rule above). So, let's consider 3 possible cases:
(q - 1) is divisible by 3
So, (q - 1)(q + 1) must also be divisible by 3
Since 4 is NOT divisible by 3, we know that (q - 1)(q + 1) - 4 is NOT divisible by 3 (from the green rule above)
If (q - 1)(q + 1) - 4 is not divisible by 3, it CANNOT be divisible by 30
q is divisible by 3
So, q² must also be divisible by 3
Since 5 is NOT divisible by 3, we know that q² - 5 is NOT divisible by 3 (from the green rule above)
If q² - 5 is not divisible by 3, then it CANNOT be divisible by 30
(q + 1) is divisible by 3
So, (q - 1)(q + 1) must also be divisible by 3
Since 4 is NOT divisible by 3, we know that (q - 1)(q + 1) - 4 is NOT divisible by 3 (from the green rule above)
If (q - 1)(q + 1) - 4 is not divisible by 3, it CANNOT be divisible by 30
Now that we've examined all 3 possible cases, we can conclude that q² - 5 CANNOT be divisible by 30
Answer = B
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Breant i liked the "rule" there, but i feel it is waay lengthy to solve the question this way. Any practical or common sense approach here?
Thanks
Thanks
You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!
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Brent@GMATPrepNow
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The only other approach I can think of (at the moment) is plugging in numbers and slowly eliminating answer choices.vinay1983 wrote:Breant i liked the "rule" there, but i feel it is waay lengthy to solve the question this way. Any practical or common sense approach here?
Thanks
q = 6: q² - 5 = 31 ELIMINATE C
q = 7: q² - 5 = 44
q = 8: q² - 5 = 59
q = 9: q² - 5 = 76 = (2)(38) ELIMINATE D
q = 10: q² - 5 = 95
q = 11: q² - 5 = 116 = (4)(29) ELIMINATE A
q = 12: q² - 5 = 139
q = 13: q² - 5 = 164 = (4)(41) ELIMINATE E
STOP
The correct answer must be B
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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GMATGuruNY
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If you don't see an elegant solution to a GMAT problem, try solving by BRUTE FORCE.[email protected] wrote:For some integer q, q^2 - 5 is divisible by all of the following EXCEPT
(A) 29
(B) 30
(C) 31
(D) 38
(E) 41
Here, any answer choice that can divide into q² - 5 can be eliminated.
To find these four incorrect answer choices, plug in increasing integer values for q:
If q=6, then q² - 5 = 31. Eliminate C.
If q=7, then q² - 5 = 44.
If q=8, then q² - 5 = 59.
If q=9, then q² - 5 = 76 = 2*38. Eliminate D.
If q=10, then q² - 5 = 95 = 5*19.
If q=11, then q² - 5 = 116 = 4*29. Eliminate A.
If q=12, then q² - 5 = 139.
If q=13, then q² - 5 = 164 = 4*41. Eliminate E.
The correct answer is B.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
