GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 11 Nov 2019, 12:23 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If x is a positive integer, is the remainder 0 when 3^x + 1

Author Message
TAGS:

### Hide Tags

Intern  Joined: 09 Feb 2011
Posts: 9
If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

2
13 00:00

Difficulty:   15% (low)

Question Stats: 78% (01:28) correct 22% (02:01) wrong based on 674 sessions

### HideShow timer Statistics

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

Originally posted by alltimeacheiver on 11 Feb 2011, 02:45.
Last edited by Bunuel on 15 Sep 2018, 05:24, edited 3 times in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 58954
Re: If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

6
13
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
_________________
##### General Discussion
Intern  Joined: 23 Mar 2009
Posts: 14
Re: If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

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.
Math Expert V
Joined: 02 Sep 2009
Posts: 58954
Re: If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

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.
_________________
Intern  Joined: 21 Jul 2012
Posts: 8
Re: If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

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?
Math Expert V
Joined: 02 Sep 2009
Posts: 58954
Re: If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

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.
_________________
Intern  Joined: 21 Jul 2012
Posts: 8
Re: If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

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 V
Joined: 02 Sep 2009
Posts: 58954
Re: If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

Bumping for review and further discussion.

For more on this kind of questions check Units digits, exponents, remainders problems collection.
_________________
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3134
Re: If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

1
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! 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
_________________
Manager  Joined: 18 Jan 2010
Posts: 238
Re: If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

1
This question checks divisibility of 3.

Concentrate only on last digit of product and not the entire value.

Now $$3^4$$= 81, i.e. last digit is 1. $$3^5$$ has last digit 3.

So $$3^{4x}$$ will have "1" as its last digit. [$$3^{4x}$$ can be written as ($$3^4$$)^x

Statement 1
$$3^{4n+2}$$ is nothing but ($$3^{4n}$$)* ($$3^2$$). The last digit of this product will be 9.

Adding 1 to this sum will yield a zero
and so the entire product will be divisible by 10.

(1) is sufficient

Statement 2

x>4.
x=5 --> This will have last digit of 3's power as 3 [($$3^4$$)*(3) => Last digit will be 3].
Add 3+1 = 4. Not divisible by 10.

x=6 --> This will have last digit of 3's power as 9 [($$3^4$$)*(9) => Last digit will be 9].
Add 9+1 = 10. Will be divisible by 10.

So Statement (B) is not sufficient.

Intern  B
Joined: 15 Aug 2018
Posts: 4
Re: If x is a positive integer, is the remainder 0 when 3^x + 1  [#permalink]

### Show Tags

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

Statement 1

Assuming n=2, x=4*2+2, x=10. Substituting in the equation 27001 is divisible by 10.

Statement 2

Assuming x=5, 3^5+1, 243+1. X=244 which is divisible by 10.

Then the answer should be ‘c’ right? Am I doing something wrong?

Posted from my mobile device Re: If x is a positive integer, is the remainder 0 when 3^x + 1   [#permalink] 15 Sep 2018, 05:22
Display posts from previous: Sort by

# If x is a positive integer, is the remainder 0 when 3^x + 1  