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If x is a positive integer, is the remainder 0 when 3^(x) +

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If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink] New post 11 Feb 2011, 01:45
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If x is a positive integer, is the remainder 0 when 3^(x) + 1 is divided by 10?

(1) x = 4n + 2, where n is a positive integer.
(2) x > 4
[Reveal] Spoiler: OA

Last edited by Bunuel on 08 Nov 2014, 04:17, edited 2 times in total.
Edited the question.
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Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink] New post 11 Feb 2011, 02:34
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alltimeacheiver wrote:
If x is a positive integer, is the remainder 0 when 3x + 1 is divided by 10?
(1) x = 4n + 2, where n is a positive integer.
(2) x > 4



Question should be as follows:

If x is a positive integer, is the remainder 0 when 3^(x) + 1 is divided by 10?

(1) x = 4n + 2, where n is a positive integer.

Last digit of \(3^x\) repeats in blocks of 4: {3, 9, 7, 1} - {3, 9, 7, 1} - ... So cyclicity of the last digit of 3 in power is 4. Now, \(3^{4n+2}\) will have the same last digit as \(3^2\) (remainder upon division 4n+2 upon cyclicity 4 is 2, which means that 3^{4n+2} will have the same last digit as 3^2). Last digit of \(3^2\) is \(9\). So \(3^{4n+2}+1\) will have the last digit \(9+1=0\). Number ending with 0 is divisible by 10 (remainder 0). Sufficient.

(2) x > 4. Clearly insufficient.

Answer: A.

Check Number Theory chapter of Math Book for more: math-number-theory-88376.html
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Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink] New post 12 Feb 2011, 12:51
If x is a positive integer, is the remainder 0 when 3x + 1 is divided by 10?
(1) x = 4n + 2, where n is a positive integer.
(2) x > 4


looking at the question, we are dealing with even numbers.
we know that x has to pos. int. we know that 3x + 1 = odd and we know that 10 is an even number. therefore remainder cannot be zero when divided by 10.

s1, x = 4n + 2.
given this eqn in s1, we know that x has to equal an even number.
When you substitute the even value of x in 3x + 1, we have an odd number such that it cannot be divisible by 10.

There s1 sufficient.

s2, x > 4. x can have a range of numbers therefore insufficient.

Ans A.
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Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink] New post 12 Feb 2011, 12:57
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maryann wrote:
If x is a positive integer, is the remainder 0 when 3x + 1 is divided by 10?
(1) x = 4n + 2, where n is a positive integer.
(2) x > 4


looking at the question, we are dealing with even numbers.
we know that x has to pos. int. we know that 3x + 1 = odd and we know that 10 is an even number. therefore remainder cannot be zero when divided by 10.

s1, x = 4n + 2.
given this eqn in s1, we know that x has to equal an even number.
When you substitute the even value of x in 3x + 1, we have an odd number such that it cannot be divisible by 10.

There s1 sufficient.

s2, x > 4. x can have a range of numbers therefore insufficient.

Ans A.


Original question is: If x is a positive integer, is the remainder 0 when [b]3^(x) + 1 is divided by 10?[/b]

Solution in my previous post.
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Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink] New post 02 Jan 2013, 01:05
Bunuel wrote:
alltimeacheiver wrote:
If x is a positive integer, is the remainder 0 when 3x + 1 is divided by 10?
(1) x = 4n + 2, where n is a positive integer.
(2) x > 4



Question should be as follows:

If x is a positive integer, is the remainder 0 when 3^(x) + 1 is divided by 10?

(1) x = 4n + 2, where n is a positive integer.

Last digit of \(3^x\) repeats in blocks of 4: {3, 9, 7, 1} - {3, 9, 7, 1} - ... So cyclicity of the last digit of 3 in power is 4. Now, \(3^{4n+2}\) will have the same last digit as \(3^2\) (remainder upon division 4n+2 upon cyclicity 4 is 2, which means that 3^{4n+2} will have the same last digit as 3^2). Last digit of \(3^2\) is \(9\). So \(3^{4n+2}+1\) will have the last digit \(9+1=0\). Number ending with 0 is divisible by 10 (remainder 0). Sufficient.

(2) x > 4. Clearly insufficient.

Answer: A.



Dear Bunuel,

I have another similar question. All info gave me are almost same. The only difference is in Statement (1): \(x= 3^n+1\). Answer goes to E. I knew that approach should be the same reference "\(4^n+2\)". I look at someone started from 1, 3,9,27,81, xxx3,xxx9,xxxx7,xxxx1, xxxx3, xxxx9, xxxx7, xxxx1, and so on.... And then reasoned that reference, which is \(4^n+2\).

I'm wondering how to quick approach that reference. Can you answer in this thread or should I submit a new post?
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Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink] New post 02 Jan 2013, 03:12
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curtis0063 wrote:
Bunuel wrote:
alltimeacheiver wrote:
If x is a positive integer, is the remainder 0 when 3x + 1 is divided by 10?
(1) x = 4n + 2, where n is a positive integer.
(2) x > 4



Question should be as follows:

If x is a positive integer, is the remainder 0 when 3^(x) + 1 is divided by 10?

(1) x = 4n + 2, where n is a positive integer.

Last digit of \(3^x\) repeats in blocks of 4: {3, 9, 7, 1} - {3, 9, 7, 1} - ... So cyclicity of the last digit of 3 in power is 4. Now, \(3^{4n+2}\) will have the same last digit as \(3^2\) (remainder upon division 4n+2 upon cyclicity 4 is 2, which means that 3^{4n+2} will have the same last digit as 3^2). Last digit of \(3^2\) is \(9\). So \(3^{4n+2}+1\) will have the last digit \(9+1=0\). Number ending with 0 is divisible by 10 (remainder 0). Sufficient.

(2) x > 4. Clearly insufficient.

Answer: A.



Dear Bunuel,

I have another similar question. All info gave me are almost same. The only difference is in Statement (1): \(x= 3^n+1\). Answer goes to E. I knew that approach should be the same reference "\(4^n+2\)". I look at someone started from 1, 3,9,27,81, xxx3,xxx9,xxxx7,xxxx1, xxxx3, xxxx9, xxxx7, xxxx1, and so on.... And then reasoned that reference, which is \(4^n+2\).

I'm wondering how to quick approach that reference. Can you answer in this thread or should I submit a new post?


Please post full question in a separate topic.
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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 ; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) ; 12. 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.


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Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink] New post 02 Jan 2013, 08:02
Bunuel wrote:

Please post full question in a separate topic.


Thank you! Link is here:
http://gmatclub.com/forum/if-x-is-a-positive-integer-is-the-remainder-0-when-3x-145117.html
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Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink] New post 09 Mar 2014, 12:21
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Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink] New post 12 May 2015, 19:14
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Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink] New post 12 May 2015, 21:02
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alltimeacheiver wrote:
If x is a positive integer, is the remainder 0 when 3^(x) + 1 is divided by 10?

(1) x = 4n + 2, where n is a positive integer.
(2) x > 4


Though this is quite an easy question, the accuracy statistics suggest that roughly 1 out of every 4 students who attempted this question got it wrong.

This may have happened because the students first straight-away started from the given statements and then, got confused in processing the question statement and the Statement 1 (and 2) together.

Here's an alternate solution that eliminates the chances of such confusion!

Let's first analyze the question statement alone:

The given expression is \(3^{x} + 1\)

The units digit of 3 can be 3 (for powers of the form 4m+1), 9 (for powers of the form 4m+2), 7 (for powers of the form 4m+3) or 1(for powers of the form 4m)

Out of these 4 possible unit digits, the sum \(3^{x} + 1\) will be divisible by 10 only when the units digit is 9.

So, the question is actually asking us to find if the power of 3, that is x, is of the form 4m + 2 or not.

Now that we've simplified the question, analyzing the 2 statements is going to be a cakewalk! :-D

Please note how the analysis in my solution is quite similar to the analysis in the solutions posted above. The point of difference comes in when I did that analysis. I did it before I went to Statements 1 and 2. The benefit of doing this analysis with the question statement itself is that I now have a very clear idea of what I need to look for, in order to answer the question. And because of this clear idea, the chances of my getting confused by irrelevant information in Statements 1 and 2 are also greatly reduced.

Hope this helped! :)

Japinder
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Re: If x is a positive integer, is the remainder 0 when 3^(x) +   [#permalink] 12 May 2015, 21:02
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