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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

Last edited by Bunuel on 12 Feb 2011, 12:57, edited 1 time in total.
Edited the question.
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Re: divisiblity with 10 [#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: divisiblity with 10 [#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: divisiblity with 10 [#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: divisiblity with 10 [#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: divisiblity with 10 [#permalink] New post 02 Jan 2013, 03:12
Expert's post
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.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

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: divisiblity with 10 [#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] 09 Mar 2014, 12:21
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