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# If neither x nor y is divisible by 3, which of the following could be

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8436
GMAT 1: 760 Q51 V42
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If neither x nor y is divisible by 3, which of the following could be  [#permalink]

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Updated on: 10 Mar 2018, 13:21
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Difficulty:

95% (hard)

Question Stats:

33% (02:40) correct 67% (02:48) wrong based on 76 sessions

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[GMAT math practice question]

x and y are integers. If neither $$x$$ nor $$y$$ is divisible by $$3$$, which of the following could be the value of $$x^2+y^2$$?

$$A. 333$$
$$B. 334$$
$$C. 335$$
$$D. 336$$
$$E. 337$$

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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Originally posted by MathRevolution on 08 Mar 2018, 02:27. Last edited by MathRevolution on 10 Mar 2018, 13:21, edited 1 time in total. ##### Most Helpful Expert Reply Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8436 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If neither x nor y is divisible by 3, which of the following could be [#permalink] ### Show Tags 11 Mar 2018, 18:32 1 4 => Consider the squares of the integers that are not divisible by 3: $$1^2 = 1, 2^2 = 4, 4^2 = 16, 5^2 = 25, 7^2 = 49, 8^2=64$$, …. They all have a remainder of $$1$$ when they are divided by $$3$$. Thus, the sum of the squares of two integers which are not divisible by $$3$$ must have a remainder of $$2$$ when it is divided by $$3$$. The only answer choice having this property is $$335$$. Therefore, C is the answer. Answer: C _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: If neither x nor y is divisible by 3, which of the following could be  [#permalink]

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08 Mar 2018, 11:18
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1
MathRevolution wrote:
[GMAT math practice question]

If neither $$x$$ nor $$y$$ is divisible by $$3$$, which of the following could be the value of $$x^2+y^2$$?

$$A. 333$$
$$B. 334$$
$$C. 335$$
$$D. 336$$
$$E. 337$$

let $$x=3k+2$$ or $$3k+1$$ and $$y=3q+2$$ or $$3q+1$$, basically a number that is not divisible by $$3$$

you can take any combination and sum the squares of the resulting number -

if $$x=3k+1$$ & $$y=3q+2$$, then $$x^2+y^2=(3k+1)^2+(3q+2)^2$$

$$=>x^2+y^2=9k^2+6k+1+9q^2+12q+4= 3(3k^2+2k+3q^2+4q)+5=3(3k^2+2k+3q^2+4q)+3+2=3(3k^2+2k+3q^2+4q+1)+2$$

Hence $$x^2+y^2$$ when divided by $$3$$ will leave a remainder of $$2$$.

Among the options only $$335$$, when divided by $$3$$ will leave a remainder of $$2$$. Hence our answer

Option C

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Essentially any number that is not divisible by 3, will leave a remainder of either 1 or 2 when divided by 3. on squaring those numbers, the remainders will also get squared to 1 or 4. so different remainder summation possible are 1+1=2, 1+4=5 & 4+4=8. Now when this number is divided by 3 it will leave a remainder of 2
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Re: If neither x nor y is divisible by 3, which of the following could be  [#permalink]

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23 Nov 2019, 05:52
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Re: If neither x nor y is divisible by 3, which of the following could be   [#permalink] 23 Nov 2019, 05:52
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