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Senior Manager  Joined: 21 Oct 2013
Posts: 408
Which of the following is NOT the sum of the squares of two  [#permalink]

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9 00:00

Difficulty:   65% (hard)

Question Stats: 61% (02:01) correct 39% (01:41) wrong based on 438 sessions

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Which of the following is NOT the sum of the squares of two integers?

(A) 36
(B) 37
(C) 65
(D) 146
(E) 147

OE
NOTE: Zero = Integer
0^2 = 0; 1^2 = 1; 2^2 = 4; 3^2 = 9; 4^2 = 16; 5^2 = 25; 6^2 = 36; 7^2 = 49; 8^2 = 64; 9^2 = 81; 10^2 = 100; 11^2 = 121; 12^2 = 144

(A): 0^2 + 6^2 = 0 + 36 = 36
(B): 1^2 + 6^2 = 1 + 36 = 37
(C): 1^2 + 8^2 = 1 + 64 = 65
(D): 5^2 + 11^2 = 25 + 121 = 146

Hi, I can see how we can find the answer. I want to know the quick way to spot combinations of (square of) 2 integers, please.

Originally posted by goodyear2013 on 09 Jan 2014, 07:54.
Last edited by Bunuel on 09 Jan 2014, 07:59, edited 1 time in total.
Edited the question.
Magoosh GMAT Instructor G
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Re: Which of the following is NOT the sum of the squares of two  [#permalink]

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goodyear2013 wrote:
Which of the following is NOT the sum of the squares of two integers?

(A) 36
(B) 37
(C) 65
(D) 146
(E) 147

Hi, I can see how we can find the answer. I want to know the quick way to spot combinations of (square of) 2 integers, please.

Dear goodyear2013,
I'm happy to respond. I don't really like this question, and I don't consider it particularly GMAT-like. On a good GMAT math question, even a hard one, there's often some perspective that allows for a particularly elegant solution. With this problem, it's easy to eliminate the first three answers, but then we have to fish around for the combination that will add up to 146 or 147. There's no particularly quick way to do this. There is nothing especially elegant about this problem. That's why I don't like it. Don't worry; you are not missing some crucial mathematical trick.

Does all this make sense?
Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Intern  Joined: 12 Sep 2012
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GMAT 1: 550 Q49 V17 Re: Which of the following is NOT the sum of the squares of two  [#permalink]

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Hi mikemcgarry,
Could you tell me why you are eliminating the first three choices. Your kind intervention is sought.
Magoosh GMAT Instructor G
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Re: Which of the following is NOT the sum of the squares of two  [#permalink]

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deya wrote:
Hi mikemcgarry,
Could you tell me why you are eliminating the first three choices. Your kind intervention is sought.

Dear deya
I'm happy to respond. We know that 6^2 = 36, so (6^2) + (0^2) = 36. (Remember, "integers" includes positive and negative and zero!!)

Then, notice that 37 = 36 + 1 = (6^2) + (1^2)
and that 65 = 64 + 1 = (8^2) + (1^2).

If you see those three things, it's easy to eliminate the first three.

Does all this make sense?
Mike _________________
Mike McGarry
Magoosh Test Prep

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Which of the following is NOT the sum of the squares of two  [#permalink]

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2
List the numbers that are smaller than the given options - 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

(A) 36 - 9+25. NO.
(B) 37 - 1+36. NO.
(C) 65 - 1+64. NO.
(D) 146 - 25+121. NO.
(E) - must be the answer.
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Re: Which of the following is NOT the sum of the squares of two  [#permalink]

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easy one, need 47 seconds to get to the answer:
(A) 36 = 25+9 or 5^2 + 3^2
(B) 37 = 36+1 or 6^2 + 1^2
(C) 65 = 49+16 or 7^2 + 4^2
(D) 146 = 121+25 or 11^2+5^2
(E) 147 - by POE, we are left with E.
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Re: Which of the following is NOT the sum of the squares of two  [#permalink]

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FireStorm wrote:
List the numbers that are smaller than the given options - 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
(A) 36 - 9+25. NO.

mvictor wrote:
easy one, need 47 seconds to get to the answer:
(A) 36 = 25+9 or 5^2 + 3^2

Dear FireStorm & mvictor,
My friends, at the risk of pointing out the obvious, 25 + 9 = 34, not 36.

Getting 36 as a sum requires remembering that zero is, in fact, an integer. Therefore, 6^2 + 0^2 = 36, a sum of the squares of integers.

My friends, never underestimate simple arithmetic. The kind of mindfulness that can keep track of each simple arithmetic on the GMAT Quant might account for a 50+ point difference in score.

Best of luck, my friends!
Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Which of the following is NOT the sum of the squares of two  [#permalink]

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Hi All,

This question has a great 'brute force' approach. Sometimes the big 'shortcut' that you'll find in a prompt is in the way that you organize your information. For this question, try writing the first 12 perfect squares VERTICALLY (instead of horizontally):

0
1
4
9
16
25
36
49
64
81
100
121
144

Looking at the numbers in this way, you can focus on the UNITS DIGITS, so it should be easier/faster to find the 4 answer choices that ARE the sum of perfect squares and the 1 that is NOT.

The first 3 answers are relatively small (and easy to spot):
36 = 0 + 36
37 = 1 + 36
65 = 1 + 64

The real work involves figuring out whether Answer D or E is the one that that you cannot get to.

If you start with the biggest number first - in this case, 144 - then there's clearly no number in the list that will get you to 146 or 147.

Next, try the 121....whatever you add to this number would need to have a 5 or a 6 as a units digit....25 is the match. Thus, you know that 146 IS possible while 147 must be the one that's not.

GMAT assassins aren't born, they're made,
Rich
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Re: Which of the following is NOT the sum of the squares of two  [#permalink]

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goodyear2013 wrote:
Which of the following is NOT the sum of the squares of two integers?

(A) 36
(B) 37
(C) 65
(D) 146
(E) 147

We can check each given answer choice to see whether it can be expressed as a sum of two squares.

(A) 36 = 6^2 + 0^2

(B) 37 = 6^2 + 1^2

(C) 65 = 8^2 + 1^2

(D) 146 = 11^2 + 5^2

Thus, we see that 147 in choice E is the only number that cannot be expressed as a sum of two squares.

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