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

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

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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.
[Reveal] Spoiler: OA

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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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New post 30 Jan 2018, 06:12
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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

Answer: A
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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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New post 30 Jan 2018, 07:39
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

Answer: A

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...

Answer is D..

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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New post 30 Jan 2018, 10:06
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

Answer: A

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...

Answer is D..

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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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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New post 06 Feb 2018, 20:20
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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Re: If x is a positive integer, what is the remainder when x is divided by [#permalink]

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New post 06 Feb 2018, 20:53
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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.

Answer: A.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
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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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New post 06 Feb 2018, 22:09
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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Re: If x is a positive integer, what is the remainder when x is divided by [#permalink]

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New post 07 Feb 2018, 09:52
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.

Answer: A.


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   [#permalink] 07 Feb 2018, 09:52
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