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Is the square of positive integer Z divisible by 9?

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Is the square of positive integer Z divisible by 9?  [#permalink]

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New post 21 Oct 2019, 17:41
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Is the square of positive integer Z divisible by 9?
(1) The sum of the digits of Z^3 is divisible by 9
(2) 3Z^4 + 16 leaves a remainder of 7 when divided by 9

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Re: Is the square of positive integer Z divisible by 9?  [#permalink]

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New post 21 Oct 2019, 20:10
Is the square of positive integer Z divisible by 9?
(1) The sum of the digits of Z^3 is divisible by 9
(2) 3Z^4 + 16 leaves a remainder of 7 when divided by 9
1) The sum of the digits of Z^3 is divisible by 9 hence z^3 is divisible by 9 . z^3 means three powers of 3 atleast . therefoew z^2 will definitely be divisible by 3. take an example 3^3 will be divisible by 9. if 111^3 is divisible by 9 then 111^2 is also divisible by 9
sufficient

2) 3Z^4 + 16 = 9l+7 simplify it
3Z^4 + 9 = 9l since 9 is divisible by 9 l that means 3Z^4 should be divisible by 9
and since z is an integer z^2 will also be divisible
sufficient
Hence answer is D
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Is the square of positive integer Z divisible by 9?  [#permalink]

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New post 21 Oct 2019, 20:34
Kinshook wrote:
Is the square of positive integer Z divisible by 9?
(1) The sum of the digits of Z^3 is divisible by 9
(2) 3Z^4 + 16 leaves a remainder of 7 when divided by 9


Analyzing the question:
For the square of Z to be divisible by 9, Z itself only needs to be divisible by 3. So we just need to find if Z has a factor of 3.

Statement 1:
This means \(Z^3\) is divisible by 9. Z must have a factor of 3 for \(Z^3\) to have a factor of 9. Hence this statement also tells us \(Z^3\) has a factor of 27. Sufficient.

Statement 2:
\(3Z^4 + 7\) has a remainder of 7 when divided by 9
-> \(3Z^3\) is divisible by 9
-> \(Z^3\) is divisible by 3.
For \(Z^3\) to have a factor of 3, Z itself must have a factor of 3. Sufficient.

Ans: D
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Re: Is the square of positive integer Z divisible by 9?  [#permalink]

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New post 28 Oct 2019, 02:38
1- Z^3 ----divisible by 9 --- so z^2 is also divisible by 9 --- Sufficient
2)3Z^4 + 16 , when divide 9 remainder comes 7 --- 16/9 --- remainder = 7 , hence 3z^4 is divisible by 9...so z^2 is also divisible by 9---sufficient
Hence D is answer

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Re: Is the square of positive integer Z divisible by 9?   [#permalink] 28 Oct 2019, 02:38
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