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

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New post Updated on: 10 Mar 2018, 13:21
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A
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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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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.
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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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New post 11 Mar 2018, 18:32
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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
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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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New post 08 Mar 2018, 11:18
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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

---------------------------------------
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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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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