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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]  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]  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.

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]  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]  12 Feb 2011, 12:57
Expert's post
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]  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.

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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Kudos [?]: 41093 [0], given: 5666

Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink]  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.

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

Please post full question in a separate topic.

http://gmatclub.com/forum/if-x-is-a-positive-integer-is-the-remainder-0-when-3x-145117.html
Math Expert
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Posts: 27228
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Kudos [?]: 41093 [0], given: 5666

Re: If x is a positive integer, is the remainder 0 when 3^(x) + [#permalink]  09 Mar 2014, 12:21
Expert's post
Bumping for review and further discussion.

For more on this kind of questions check Units digits, exponents, remainders problems collection.
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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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