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

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

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.

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.

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

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.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

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.

Answer: E

Hi EMPOWERgmatRichC , why did you choose to write only the first 12 perfect squares?

Hi rrn98,

Looking at the answer choices, the largest value (147) is just a little bigger than 12^2 = 144 - and since we're looking for the SUMS of TWO perfect squares, anything larger than 12^2 would be too big.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at: Rich.C@empowergmat.com

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