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# If x is a positive integer, what is the remainder when x is divided by

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Math Expert
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If x is a positive integer, what is the remainder when x is divided by  [#permalink]

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30 Jan 2018, 06:59
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95% (hard)

Question Stats:

24% (01:39) correct 76% (01:35) wrong based on 142 sessions

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Fresh GMAT Club Tests' Question

If x is a positive integer, what is the remainder when x is divided by 2?

(1) $$(-1)^{(x^2)} = -1$$

(2) $$n^x = n^{(2x - 1)}$$

OFFICIAL SOLUTION IS HERE.

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Re: If x is a positive integer, what is the remainder when x is divided by  [#permalink]

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30 Jan 2018, 07:12
1
2
St1: (-1)^odd = -1 --> x^2 = odd --> x = odd
odd/2 --> Rem = 1
Sufficient

St2: n^x = n^(2x - 1)
If n = 1; x can be odd or even.
Not sufficient

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Re: If x is a positive integer, what is the remainder when x is divided by  [#permalink]

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30 Jan 2018, 08:39
Vyshak wrote:
St1: (-1)^odd = -1 --> x^2 = odd --> x = odd
odd/2 --> Rem = 1
Sufficient

St2: n^x = n^(2x - 1)
If n = 1; x can be odd or even.
Not sufficient

I think its D ..

Statemnt 1 is right as u said.
But in st.2

As base are same so we can equalise expo.
x= 2x-1
x=1
So sufficient...

Correct me if i am wrong...

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Re: If x is a positive integer, what is the remainder when x is divided by  [#permalink]

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30 Jan 2018, 11:06
viv007 wrote:
Vyshak wrote:
St1: (-1)^odd = -1 --> x^2 = odd --> x = odd
odd/2 --> Rem = 1
Sufficient

St2: n^x = n^(2x - 1)
If n = 1; x can be odd or even.
Not sufficient

I think its D ..

Statemnt 1 is right as u said.
But in st.2

As base are same so we can equalise expo.
x= 2x-1
x=1
So sufficient...

Correct me if i am wrong...

Sent from my BND-AL10 using GMAT Club Forum mobile app

Hi

I think we cannot always equate the exponents if we dont know anything about the base. What if n = -1.
(-1)^3 is same as (-1)^5, but that doesn't mean 3 is equal to 5
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Re: If x is a positive integer, what is the remainder when x is divided by  [#permalink]

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05 Feb 2018, 12:40
1
Bunuel wrote:

Fresh GMAT Club Tests' Question

If x is a positive integer, what is the remainder when x is divided by 2?

(1) $$(-1)^{(x^2)} = -1$$

(2) $$n^x = n^{(2x - 1)}$$

I think the answer must be D.

Statement 1, for (−1)^(x^2)=−1, x^2 must be an odd number and odd number when divided by 2 always give a remainder 1. Hence sufficient.

Statement 2, considering n=-1, (-1)^x = (-1)^(2x-1) only when both the exponents are either odd or even. When x is even, 2x-1 cannot be even. Therefore x is odd and the remainder again is 1. Hence sufficient.

Correct me if I am wrong.
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Re: If x is a positive integer, what is the remainder when x is divided by  [#permalink]

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06 Feb 2018, 21:20
I think ur right coz when I take X=2 then 2x-1 comes to be odd. When I take X as odd 2x-1 is odd. So second option is correct too
Ans Is D

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Posts: 50007
Re: If x is a positive integer, what is the remainder when x is divided by  [#permalink]

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06 Feb 2018, 21:53
OFFICIAL SOLUTION:

If x is a positive integer, what is the remainder when x is divided by 2?

(1) $$(-1)^{(x^2)} = -1$$.

This implies that x^2 is odd (if it were even, then (-1)^even = 1). x^2 = odd, on the other hand means that x is odd (since given that x is an integer). Any odd number divided by 2 gives the remainder of 1. Sufficient.

(2) $$n^x = n^{(2x - 1)}$$.

Be careful not to fall into the trap. Remember we can automatically equate the exponents of equal bases when that base does not equal 0, 1 or -1:

$$1^x = 1^y$$, for any values of x and y (they are not necessarily equal);
$$(-1)^x = (-1)^y$$, for any even values of x and y (they are not necessarily equal);
$$0^x = 0^y$$, for any non-zero x and y (they are not necessarily equal).

Thus, for $$n^x = n^{(2x - 1)}$$, we could equate the exponents if we knew that n is not 0, 1 or -1. In this case we'd have: x = 2x - 1, which would give x = 1. But if n is 0, 1 or -1, then we cannot equate the exponents. For example, if n = 1, then x can be even as well as odd. Not sufficient.

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Re: If x is a positive integer, what is the remainder when x is divided by  [#permalink]

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06 Feb 2018, 23:09
Thanks for the explanation. Understand the key take away here is that exponents can be equated only when base is not equal to 0, 1 or -1.

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Posts: 96
Re: If x is a positive integer, what is the remainder when x is divided by  [#permalink]

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07 Feb 2018, 10:52
Bunuel wrote:
OFFICIAL SOLUTION:

If x is a positive integer, what is the remainder when x is divided by 2?

(1) $$(-1)^{(x^2)} = -1$$.

This implies that x^2 is odd (if it were even, then (-1)^even = 1). x^2 = odd, on the other hand means that x is odd (since given that x is an integer). Any odd number divided by 2 gives the remainder of 1. Sufficient.

(2) $$n^x = n^{(2x - 1)}$$.

Be careful not to fall into the trap. Remember we can automatically equate the exponents of equal bases when that base does not equal 0, 1 or -1:

$$1^x = 1^y$$, for any values of x and y (they are not necessarily equal);
$$(-1)^x = (-1)^y$$, for any even values of x and y (they are not necessarily equal);
$$0^x = 0^y$$, for any non-zero x and y (they are not necessarily equal).

Thus, for $$n^x = n^{(2x - 1)}$$, we could equate the exponents if we knew that n is not 0, 1 or -1. In this case we'd have: x = 2x - 1, which would give x = 1. But if n is 0, 1 or -1, then we cannot equate the exponents. For example, if n = 1, then x can be even as well as odd. Not sufficient.

Thank you "BUNUEL"...for nice expalination...

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Re: If x is a positive integer, what is the remainder when x is divided by &nbs [#permalink] 07 Feb 2018, 10:52
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