It is currently 28 Jun 2017, 12:43

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If r, s, and t are all positive integers, what is the

Author Message
TAGS:

### Hide Tags

Intern
Joined: 28 Jul 2011
Posts: 3
Concentration: Entrepreneurship, Organizational Behavior
GMAT Date: 10-18-2012
If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

02 Aug 2012, 16:00
13
This post was
BOOKMARKED
00:00

Difficulty:

35% (medium)

Question Stats:

64% (01:57) correct 36% (00:57) wrong based on 569 sessions

### HideShow timer Statistics

If r, s, and t are all positive integers, what is the remainder when 2^(rst) is divided by 10?

(1) s is even
(2) rs = 4
[Reveal] Spoiler: OA

Last edited by Bunuel on 02 Aug 2012, 16:10, edited 1 time in total.
Moved to DS subforum, edited the question and renamed the topic.
Math Expert
Joined: 02 Sep 2009
Posts: 39751
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

02 Aug 2012, 16:19
2
KUDOS
Expert's post
7
This post was
BOOKMARKED
If r, s, and t are all positive integers, what is the remainder when 2^(rst) is divided by 10?

First of all, when a positive integer is divided by 10, the remainder is the units digit of that integer. For example, 30 divided by 10 yields the remainder of 0, 31 divided by 10 yields the remainder of 1, 32 divided by 10 yields the remainder of 2, ...

Next, the units digit of 2 in positive integer power repeats in blocks of 4: {2, 4, 8, 6}

The units digit of 2^1 is 2;
The units digit of 2^2 is 4;
The units digit of 2^3 is 8;
The units digit of 2^4 is 6;
The units digit of 2^5 is 2, AGAIN;
...

(1) s is even --> rst is even, hence the units digit of 2^(rst) is either 4 or 6. Not sufficient.

(2) rs = 4 --> rst is a multiple of 4, hence the units digit of 2^(rst) is the same as the units digit of 2^4 so 6, which means that the remainder upon division of 2^(rst) by 10 is 6. Sufficient.

_________________
Director
Joined: 03 Aug 2012
Posts: 894
Concentration: General Management, General Management
GMAT 1: 630 Q47 V29
GMAT 2: 680 Q50 V32
GPA: 3.7
WE: Information Technology (Investment Banking)
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

11 Aug 2013, 05:39
1
KUDOS
r,s,t are +ve

REM(2^rst/10) ?

(1).

s is even also even * even = even and even*odd=even

But REM(2^2/10) and REM(2^4/10) are different hence insufficient .

(2).

rs=4

REM(2^4t/10)

REM(2^4/10) ....REM(2^8/10).......REM(2^12/10) .... All are same

Hence sufficient

(B). it is !
_________________

Rgds,
TGC!
_____________________________________________________________________
I Assisted You => KUDOS Please
_____________________________________________________________________________

Intern
Joined: 02 Jan 2013
Posts: 2
Location: India
WE: Programming (Computer Software)
Re: If r, s, and t are all positive integers, what is the remain [#permalink]

### Show Tags

19 Aug 2013, 06:42
1
KUDOS
a) if s is even, i.e. rst = even -> 2^even/10 -> can't determine
b) rs = 4, i.e. rst = 4t -> 2^4t/10 -> 2^4t will always have 6 in unit's place(always the multiplication for unit place will be 6*6), so remainder will be 6 -> determined.

Director
Joined: 14 Dec 2012
Posts: 832
Location: India
Concentration: General Management, Operations
GMAT 1: 700 Q50 V34
GPA: 3.6
Re: If r, s, and t are all positive integers, what is the remain [#permalink]

### Show Tags

19 Aug 2013, 06:43
Stiv wrote:
If r, s, and t are all positive integers, what is the remainder when $$2^{rst}$$ is divided by 10?

(1) s is even

(2) rs = 4

since we are dividing by $$10$$ means remainder will depend on only the unit digit of $$2^{rst}$$

moreover we know unit digit of$$2^{4n} = 6$$
unit digit of $$2^{4n+1} = 2$$
unit digit of $$2^{4n+2} = 4$$
unit digit of $$2^{4n+3} = 8$$

so determining the unit digit we should be able to make $$2^{rst}$$ in any of the above form.

(1) s is even
not clear it can be of form $$2^{4n} or 2^{4n+2}$$
hence not sufficient.

2)$$rs = 4$$
clearly we will get $$2^{4s}$$==>hence unit digit will be$$6$$
hence remainder will be $$6$$.
hence sufficient.

hence B
_________________

When you want to succeed as bad as you want to breathe ...then you will be successfull....

GIVE VALUE TO OFFICIAL QUESTIONS...

learn AWA writing techniques while watching video : http://www.gmatprepnow.com/module/gmat-analytical-writing-assessment

Math Expert
Joined: 02 Sep 2009
Posts: 39751
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

09 Mar 2014, 13:22
Bumping for review and further discussion.

For more on this kind of questions check Units digits, exponents, remainders problems collection.
_________________
Manager
Joined: 01 May 2013
Posts: 62
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

09 Mar 2014, 16:40
Any integer that does not end in 0 will have a positive remainder when divided by 10. Specifically, the remainder will be equal to the ones column. No power of 2 ends in 0. We need the units digit of 2^(rst).

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64

Every fourth power of 2 repeats.

Statement 1 tells us the units digit will be 4 or 6. Not sufficient.

Statement 2 is sufficient. rst will be a multiple of 4, with units digit 6. Sufficient.

Intern
Joined: 29 Sep 2012
Posts: 13
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

12 Jun 2014, 20:11
If r, s, and t are all positive integers, what is the remainder of 2^p/10, if p = rst?

(1) s is even

(2) p = 4t

Hi everyone, i have a doubt with statement B. since p=4t so when divided by 10 we can cancel a 2 from both numerator and denominator so we have 2^3t/5 which is 8^t/5 so in this case we have different remainders each time.

Math Expert
Joined: 02 Sep 2009
Posts: 39751
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

13 Jun 2014, 01:37
snehamd1309 wrote:
If r, s, and t are all positive integers, what is the remainder of 2^p/10, if p = rst?

(1) s is even

(2) p = 4t

Hi everyone, i have a doubt with statement B. since p=4t so when divided by 10 we can cancel a 2 from both numerator and denominator so we have 2^3t/5 which is 8^t/5 so in this case we have different remainders each time.

$$\frac{2^{4t}}{10}=\frac{2^{4t-1}}{5}$$ not 2^3t/5. Also, when we are asked to find the remainder of a/b it's not correct to reduce the fraction and find the remainder of the resulting fraction. For example, the remainder when 15 is divided by 6 is 3, but if you reduce that by 3 and find the remainder of 5 by 2 you'd get the remainder of 1.

Hope its clear.
_________________
Intern
Joined: 29 Sep 2012
Posts: 13
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

13 Jun 2014, 03:19
Bunuel wrote:
snehamd1309 wrote:
If r, s, and t are all positive integers, what is the remainder of 2^p/10, if p = rst?

(1) s is even

(2) p = 4t

Hi everyone, i have a doubt with statement B. since p=4t so when divided by 10 we can cancel a 2 from both numerator and denominator so we have 2^3t/5 which is 8^t/5 so in this case we have different remainders each time.

$$\frac{2^{4t}}{10}=\frac{2^{4t-1}}{5}$$ not 2^3t/5. Also, when we are asked to find the remainder of a/b it's not correct to reduce the fraction and find the remainder of the resulting fraction. For example, the remainder when 15 is divided by 6 is 3, but if you reduce that by 3 and find the remainder of 5 by 2 you'd get the remainder of 1.

Hope its clear.

Thanks Bunuel for your reply. I understood that one should not cancel out however cant understand 2^4t/10 is simplified into 2^4t-1/5 and not 2^3t/5. Don't we cancel the powers. for Example 2^3/2= 2^2. then why cant it be in the previous one.Please help.Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 39751
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

13 Jun 2014, 03:27
snehamd1309 wrote:
Bunuel wrote:
snehamd1309 wrote:
If r, s, and t are all positive integers, what is the remainder of 2^p/10, if p = rst?

(1) s is even

(2) p = 4t

Hi everyone, i have a doubt with statement B. since p=4t so when divided by 10 we can cancel a 2 from both numerator and denominator so we have 2^3t/5 which is 8^t/5 so in this case we have different remainders each time.

$$\frac{2^{4t}}{10}=\frac{2^{4t-1}}{5}$$ not 2^3t/5. Also, when we are asked to find the remainder of a/b it's not correct to reduce the fraction and find the remainder of the resulting fraction. For example, the remainder when 15 is divided by 6 is 3, but if you reduce that by 3 and find the remainder of 5 by 2 you'd get the remainder of 1.

Hope its clear.

Thanks Bunuel for your reply. I understood that one should not cancel out however cant understand 2^4t/10 is simplified into 2^4t-1/5 and not 2^3t/5. Don't we cancel the powers. for Example 2^3/2= 2^2. then why cant it be in the previous one.Please help.Thanks

$$\frac{a^n}{a^m}=a^{n-m}$$. Hence, $$\frac{2^3}{2^2}=2^{3-2}=2$$ the same way: $$\frac{2^{4t}}{2}=2^{4t-1}$$.

Theory on Exponents: math-number-theory-88376.html

All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Hope this helps.
_________________
Manager
Joined: 20 Dec 2013
Posts: 132
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

13 Jun 2014, 04:13
jcmorales2012 wrote:
If r, s, and t are all positive integers, what is the remainder when 2^(rst) is divided by 10?

(1) s is even
(2) rs = 4

Remainder when divided by 10 = last digit

Last digit of 2 ^4 = 2^8 = 2^16 and so on.....

Statement I in insufficient:

If rst = 2 then 2 ^2 = 4 and rst =4 then 2 ^4 = last digit is 6

Statement II is sufficient:

If rs = 4 then rst is a multiple of 4 which means rst = 4k hence 2^4, 2^8, 2^12 will give the same last digit.

_________________

76000 Subscribers, 7 million minutes of learning delivered and 5.6 million video views

Perfect Scores
http://perfectscores.org

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16024
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

02 Jul 2015, 20:23
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16024
Re: If r, s, and t are all positive integers, what is the [#permalink]

### Show Tags

16 Nov 2016, 01:29
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If r, s, and t are all positive integers, what is the   [#permalink] 16 Nov 2016, 01:29
Similar topics Replies Last post
Similar
Topics:
If r is a positive integer, what is the value of r ? 1 01 Aug 2016, 06:30
7 If r, s, and t are all positive integers, what is the remainder of 2^p 3 05 Jul 2016, 02:58
2 What is the product of positive integers r and s ? 2 29 Mar 2017, 05:56
26 Given that R is positive three-digit integer, what is the 16 01 May 2017, 08:23
69 What is the tens digit of the positive integer r? 25 19 Jan 2017, 05:50
Display posts from previous: Sort by