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For a nonnegative integer n, if the remainder is 1 when 2^n

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For a nonnegative integer n, if the remainder is 1 when 2^n [#permalink] New post 05 Nov 2010, 19:38
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For a nonnegative integer n, if the remainder is 1 when 2^n is divided by 3, then which of the following must be true?

I. n is greater than zero.
II. 3^n = (-3)^n
III. √2^n is an integer.

A. I only
B. II only
C. I and II
D. I and III
E. II and III
[Reveal] Spoiler: OA

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Last edited by Bunuel on 18 Jun 2013, 13:07, edited 1 time in total.
Edited the question.
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Re: For a nonnegative integer n [#permalink] New post 05 Nov 2010, 20:32
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monirjewel wrote:
For a nonnegative integer n, if the remainder is 1 when 2^n is divided by 3, then which of the following must be true?
I. n is greater than zero.
II. 3^n = (-3)^n
III. √2^n is an integer.
A. I only
B. II only
C. I and II
D. I and III
E. II and III


Please check the questions when posting.

Given: n=integer\geq{0} and 2^n=3q+1, for some non-negative integer q:

If n=0=even --> 2^0=1 --> remainder upon division 1 by 3 is 1 - OK;
If n=1=odd --> 2^1=2 --> remainder upon division 2 by 3 is 2 - not OK;
If n=2=even --> 2^2=4 --> remainder upon division 4 by 3 is 1 - OK;
If n=3=odd --> 2^3=8 --> remainder upon division 8 by 3 is 2 - not OK;
...

So we can see the pattern of reminders 1-2-1-2-.... --> given condition that the remainder is 1 when 2^n is divided by 3 holds true when n=even. So n must be non-negative even number: 0, 2, 4, ...

I. n is greater than zero --> not necessarily true, as n can be zero;
II. 3^n = (-3)^n --> as n is even then this statement is always true;
III. \sqrt{2}^n=integer --> as n is non-negative even number then this statement is always true.

Answer: E (II and III only).
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Re: For a nonnegative integer n [#permalink] New post 16 Feb 2011, 15:58
Thank you, Bunuel! I checked a lot of sources, but your explanation is the best, as usual!=)
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Re: For a nonnegative integer n [#permalink] New post 16 Mar 2012, 03:17
Bunuel, Your explanation is always the BEST! THANK U.......
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Re: For a nonnegative integer For a nonnegative integer n, if [#permalink] New post 21 Mar 2012, 20:09
just a question on this is it (√2)^n or (√2^n).

Just have some confusion on this.
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Re: For a nonnegative integer For a nonnegative integer n, if [#permalink] New post 22 Mar 2012, 00:08
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Re: For a nonnegative integer n, if the remainder is 1 when 2n [#permalink] New post 26 Mar 2012, 01:31
Tried solving using the 1st post and got mightily confused....Thx Bunuel!!
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Re: For a nonnegative integer n, if the remainder is 1 when 2n [#permalink] New post 06 Jun 2013, 05:32
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Re: For a nonnegative integer n [#permalink] New post 06 Jun 2013, 09:19
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Bunuel wrote:
monirjewel wrote:
For a nonnegative integer n, if the remainder is 1 when 2^n is divided by 3, then which of the following must be true?
I. n is greater than zero.
II. 3^n = (-3)^n
III. √2^n is an integer.
A. I only
B. II only
C. I and II
D. I and III
E. II and III


We know that x^n-a^n is divisible by (x-a) AND (x+a) ONLY when n is an even integer.

Now , from the given problem, 2^n = 3q+1--> \frac{2^n-1^n}{3} = q(An integer)--> \frac{2^n-1^n}{(2+1)} = q.
Thus, n HAS to be even. Now, only options II and III stand for n= even.
E.
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Re: For a nonnegative integer n, if the remainder is 1 when 2n [#permalink] New post 18 Jun 2013, 13:03
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Question is not typed clearly...solved it as 2n and not 2^n :(
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Re: For a nonnegative integer n, if the remainder is 1 when 2n [#permalink] New post 18 Jun 2013, 13:08
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Re: For a nonnegative integer n [#permalink] New post 28 Jun 2013, 08:34
Bunuel wrote:
monirjewel wrote:
For a nonnegative integer n, if the remainder is 1 when 2^n is divided by 3, then which of the following must be true?
I. n is greater than zero.
II. 3^n = (-3)^n
III. √2^n is an integer.
A. I only
B. II only
C. I and II
D. I and III
E. II and III


Please check the questions when posting.

Given: n=integer\geq{0} and 2^n=3q+1, for some non-negative integer q:

If n=0=even --> 2^0=1 --> remainder upon division 1 by 3 is 1 - OK;
If n=1=odd --> [color=#ff0000][b]2^2=2 [/color][/b]--> remainder upon division 2 by 3 is 2 - not OK;
If n=2=even --> 2^2=4 --> remainder upon division 4 by 3 is 1 - OK;
If n=3=odd --> 2^3=8 --> remainder upon division 8 by 3 is 2 - not OK;
...

So we can see the pattern of reminders 1-2-1-2-.... --> given condition that the remainder is 1 when 2^n is divided by 3 holds true when n=even. So n must be non-negative even number: 0, 2, 4, ...

I. n is greater than zero --> not necessarily true, as n can be zero;
II. 3^n = (-3)^n --> as n is even then this statement is always true;
III. \sqrt{2}^n=integer --> as n is non-negative even number then this statement is always true.

Answer: E (II and III only).


Dont you think this should be 2^1= 2 and not 2^2=2???
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Re: For a nonnegative integer n [#permalink] New post 28 Jun 2013, 08:39
Expert's post
prateekbhatt wrote:
Bunuel wrote:
monirjewel wrote:
For a nonnegative integer n, if the remainder is 1 when 2^n is divided by 3, then which of the following must be true?
I. n is greater than zero.
II. 3^n = (-3)^n
III. √2^n is an integer.
A. I only
B. II only
C. I and II
D. I and III
E. II and III


Please check the questions when posting.

Given: n=integer\geq{0} and 2^n=3q+1, for some non-negative integer q:

If n=0=even --> 2^0=1 --> remainder upon division 1 by 3 is 1 - OK;
If n=1=odd --> [color=#ff0000][b]2^2=2 [/color][/b]--> remainder upon division 2 by 3 is 2 - not OK;
If n=2=even --> 2^2=4 --> remainder upon division 4 by 3 is 1 - OK;
If n=3=odd --> 2^3=8 --> remainder upon division 8 by 3 is 2 - not OK;
...

So we can see the pattern of reminders 1-2-1-2-.... --> given condition that the remainder is 1 when 2^n is divided by 3 holds true when n=even. So n must be non-negative even number: 0, 2, 4, ...

I. n is greater than zero --> not necessarily true, as n can be zero;
II. 3^n = (-3)^n --> as n is even then this statement is always true;
III. \sqrt{2}^n=integer --> as n is non-negative even number then this statement is always true.

Answer: E (II and III only).


Dont you think this should be 2^1= 2 and not 2^2=2???


Yes. Typo edited. Thank you.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

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.


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Re: For a nonnegative integer n [#permalink] New post 28 Jun 2013, 09:12
Bunuel wrote:

Dont you think this should be 2^1= 2 and not 2^2=2???


Yes. Typo edited. Thank you.[/quote]

\sqrt{2}^0 equals 1 or \sqrt{2} its self
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Re: For a nonnegative integer n [#permalink] New post 28 Jun 2013, 09:14
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Re: For a nonnegative integer n, if the remainder is 1 when 2^n [#permalink] New post 27 Oct 2014, 22:20
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Re: For a nonnegative integer n, if the remainder is 1 when 2^n   [#permalink] 27 Oct 2014, 22:20
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