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

It is currently 20 Aug 2018, 21:10

Close

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
Your Progress

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.

Close

Request Expert Reply

Confirm Cancel

If x is a positive integer, what is the units digit of

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

VP
VP
User avatar
Joined: 25 Nov 2004
Posts: 1450
If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 11 Jul 2006, 10:32
5
31
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

67% (01:54) correct 33% (01:58) wrong based on 781 sessions

HideShow timer Statistics

If x is a positive integer, what is the units digit of \((24)^{(2x + 1)}*(33)^{(x + 1)}*(17)^{(x + 2)}*(9)^{(2x)}\) ?

A. 4
B. 6
C. 7
D. 8
E. 9
Most Helpful Community Reply
Current Student
User avatar
B
Joined: 29 Mar 2012
Posts: 317
Location: India
GMAT 1: 640 Q50 V26
GMAT 2: 660 Q50 V28
GMAT 3: 730 Q50 V38
GMAT ToolKit User
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 30 Jun 2012, 23:45
12
10
Hi,

\((24)^{2x + 1}*(33)^{x + 1}*(17)^{x + 2}*(9)^{2x}\)
\(=(24^2)^x*24*33^x*33*17^x*17^2*(9^2)^x\)
\(=(24^2)^x*(33*17)^x*(24*33)*17^2*(9^2)^x\)
considering only the unit digits;
\(=(6)^x*(1)^x*2*9*(1)^x\)
\(=6*1*2*9*1\)
\(=8\)

Answer (D)

Regards,
General Discussion
Manager
Manager
User avatar
Joined: 24 Oct 2005
Posts: 164
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 11 Jul 2006, 11:26
The answer is D

(24)^(2x + 1)
(2x+1) is an odd number since even + odd = odd
4 to an even power ends with a 4.

(33)^(x + 1)*(17)^(x + 2)
x could be even or odd. So if (x+1) is even, (x+2) will be odd. On the contrary, if (x+1) is odd, (x+2) will be even.
Trying some values for x:

If x =1

(33)^(2)*(17)^(3) = 9*3 = 27

If x= 2

(33)^(3)*(17)^(4) = 7*1 = 7

and so on......the table below shows the pattern:

Number
3 7
Power
2 9 9
3 7 3
4 1 1
5 3 7
6 9 9


(9)^(2x)

the power will be even, thus the units digit is 1.



Multiplying 4*7*1 = 8
Senior Manager
Senior Manager
User avatar
Joined: 07 Jul 2005
Posts: 399
Location: Sunnyvale, CA
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 11 Jul 2006, 11:44
2
1
MA wrote:
If x is a positive integer, what is the units digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)?

(A) 4
(B) 6
(C) 7
(D) 8
(E) 9

In these questions, since the answer will be true for any value of x, we can choose the min. value of x (in this case 1) and solve..

(D)
GMAT Instructor
avatar
B
Joined: 04 Jul 2006
Posts: 1253
Location: Madrid
CAT Tests
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 11 Jul 2006, 14:14
4
3
Remember that in the GMAT, it pays to look for shortcuts!

33^(x+1)*17^(x+2)= ((33*17)^(x+1))*17 which has a units digit of 7!
VP
VP
User avatar
Joined: 25 Nov 2004
Posts: 1450
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post Updated on: 11 Jul 2006, 20:31
good one. thanx kevin..


kevincan wrote:
Remember that in the GMAT, it pays to look for shortcuts!


33^(x+1)*17^(x+2)= ((33*17)^(x+1))*17 which has a units digit of 7!


= (24)^(2x + 1) * (33)^(x + 1) * (17)^(x + 2) * (9)^(2x)
= 24 (336^x) * 33 (33^x) * 289 (17^x) * (81^x)
= (24 x 33 x 289) (336 x 33 x 17 x 81)^x

(24 x 33 x 289) has unit digit of 8
(336 x 33 x 17 x 81)^x has unit digit f 6 irrspective of value of x.
so (24 x 33 x 289) (336 x 33 x 17 x 81)^x has unit digit of 8.

Originally posted by MA on 11 Jul 2006, 19:39.
Last edited by MA on 11 Jul 2006, 20:31, edited 2 times in total.
Intern
Intern
avatar
Joined: 25 Mar 2012
Posts: 24
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 13 Jul 2012, 09:40
3
solved in 00:34

consider x =1
24^3 * 33^2 * 17^3 * 9^2

units digit will be
4*9*3*1

=> 8

satisfies for any positive integer.
Intern
Intern
avatar
Joined: 23 Jul 2013
Posts: 20
GMAT ToolKit User
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 11 Sep 2013, 00:32
1
2
Let value of x;

x=1

24^3*33^2*17^3*9^2

unit digits = 4*9*3*1 = 108

unit digit = 8

answer = D
Intern
Intern
avatar
Joined: 29 Sep 2012
Posts: 12
GMAT ToolKit User
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 13 Jun 2014, 20:39
Bunuel wrote:
Merging similar topics.


Hi Bunuel,

In these kind of questions where in it is asked that X is a positive integer, is substituting and value of X a good idea to solve it quickly. Though by taking X=1 or 2 i have arrived at unit's digit as 8 but will it hold for all values of X.

Thanks
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 48068
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 14 Jun 2014, 01:33
snehamd1309 wrote:
Bunuel wrote:
Merging similar topics.


Hi Bunuel,

In these kind of questions where in it is asked that X is a positive integer, is substituting and value of X a good idea to solve it quickly. Though by taking X=1 or 2 i have arrived at unit's digit as 8 but will it hold for all values of X.

Thanks


Yes. There is only one correct answer in a PS question, thus every x should give the same correct answer.

Units digits, exponents, remainders problems directory: new-units-digits-exponents-remainders-problems-168569.html

Hope it helps.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | 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.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Intern
Intern
avatar
Joined: 28 Mar 2014
Posts: 22
Location: India
GPA: 3
WE: Business Development (Retail Banking)
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 14 Jun 2014, 06:24
24^2x+1 for this 2x+1 is odd therefore the unit digit is 4.
33^x+1 can be clubbed with 17^x+2 which means 17^(x+2) can be written as 17^(x+1)*17. So we can write (33*17)^(x+1)*17. 33*17 gives unit digit as 1. Therefore we can write 1^(x+1)*17 = unit digit as 7.
9^2x gives unit digit as 1 since 9 is raised to an even no. of power.
so, the total equation becomes 4*1*7*1 = unit digit as 8.
Intern
Intern
avatar
Joined: 07 Mar 2014
Posts: 20
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 05 Jun 2015, 01:35
Bunuel wrote:
snehamd1309 wrote:
Bunuel wrote:
Merging similar topics.


Hi Bunuel,

In these kind of questions where in it is asked that X is a positive integer, is substituting and value of X a good idea to solve it quickly. Though by taking X=1 or 2 i have arrived at unit's digit as 8 but will it hold for all values of X.

Thanks


Yes. There is only one correct answer in a PS question, thus every x should give the same correct answer.

Units digits, exponents, remainders problems directory: new-units-digits-exponents-remainders-problems-168569.html

Hope it helps.



Is my logic right ( i tried to plug some numbers and get to validate my logic. But i am not sure if this logic can be generalized)

(24)^(2x + 1) (33)^(x + 1) (17)^(x + 2) (9)^(2x)

this can be written in terms of unit digits as :
(4)^(2x + 1) (3)^(x + 1) (7)^(x + 2) (9)^(2x)

Then
(4)^(2x + 1) : give unit digit 4
(3)^(x + 1) : gives unit digit based 3^x and 3
(7)^(x + 2) : give unit digit based 7^x+2 and 7^2 --> 7^2 has unit digit 9
(9)^(2x) : give unit digit 1

therefore the terms can be reduced to :(in unit digits)
4* 3* 3^x * 7^x * 9 * 1
= 2*9 * 3^x *7^x
= 8 * 3^x *7^x
= 8 * (21)^x ( can i combine the two unit digits 3^ x * 7^ x = (21)^x = 1^x)


= 8 * 1^ x = unit digit = 8

my only doubt can i generalize this logic : 3^ x * 7^ x = (21)^x = 1^x to all unit integers.
SVP
SVP
User avatar
P
Joined: 08 Jul 2010
Posts: 2139
Location: India
GMAT: INSIGHT
WE: Education (Education)
Reviews Badge
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 05 Jun 2015, 04:41
2
Jam2014 wrote:
Bunuel wrote:
snehamd1309 wrote:
Merging similar topics.


Hi Bunuel,

In these kind of questions where in it is asked that X is a positive integer, is substituting and value of X a good idea to solve it quickly. Though by taking X=1 or 2 i have arrived at unit's digit as 8 but will it hold for all values of X.

Thanks

Yes. There is only one correct answer in a PS question, thus every x should give the same correct answer.

Units digits, exponents, remainders problems directory: new-units-digits-exponents-remainders-problems-168569.html

Hope it helps.



Is my logic right ( i tried to plug some numbers and get to validate my logic. But i am not sure if this logic can be generalized)

(24)^(2x + 1) (33)^(x + 1) (17)^(x + 2) (9)^(2x)

this can be written in terms of unit digits as :
(4)^(2x + 1) (3)^(x + 1) (7)^(x + 2) (9)^(2x)

Then
(4)^(2x + 1) : give unit digit 4
(3)^(x + 1) : gives unit digit based 3^x and 3
(7)^(x + 2) : give unit digit based 7^x+2 and 7^2 --> 7^2 has unit digit 9
(9)^(2x) : give unit digit 1

therefore the terms can be reduced to :(in unit digits)
4* 3* 3^x * 7^x * 9 * 1
= 2*9 * 3^x *7^x
= 8 * 3^x *7^x
= 8 * (21)^x ( can i combine the two unit digits 3^ x * 7^ x = (21)^x = 1^x)


= 8 * 1^ x = unit digit = 8

my only doubt can i generalize this logic : 3^ x * 7^ x = (21)^x = 1^x to all unit integers.


Hi jam,

Yes, We can generalize this principle for all the Integers with more than 1 digit only

Reason: In calculation of the Unit Digit, only Unit Digit matters and all the digits other than unit digit of numbers become redundant.

i.e. \((857)^x\) will have same Unit Digit as \(7^x\)

I hope clears the doubts!!!
_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

SVP
SVP
User avatar
P
Joined: 08 Jul 2010
Posts: 2139
Location: India
GMAT: INSIGHT
WE: Education (Education)
Reviews Badge
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 05 Jun 2015, 04:54
Quote:
If x is a positive integer, what is the units digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)?

(A) 4
(B) 6
(C) 7
(D) 8
(E) 9


Here is another method to answer this question very quickly

Just observe the Language of the question "If x is a positive integer, what is the units digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)?"

The "is" part confirms that the result of this question will be unique for any value of x which is a positive Integer.

Hence this question becomes much easier for any chosen positive integer value of x,

Let's take x = 1

Now the question becomes

(24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x) = (24)^(2 + 1)*(33)^(1 + 1)*(17)^(1 + 2)*(9)^(2)

IMPORTANT POINT : In calculation of the Unit Digit, only Unit Digit matters and all the digits other than unit digit of numbers become redundant.

But (24)^(2 + 1) will have same unit digit as 4^(2+1) i.e. 4^3 i.e. 4

and But (33)^(1 + 1) will have same unit digit as 3^(1+1) i.e. 3^2 i.e. 9

and But (17)^(1 + 2) will have same unit digit as 7^(1+2) i.e. 7^3 i.e. 3

and But (9)^(2) will be 1

i.e. Unit digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x) = 4 x 9 x 3 x 1 = 8

Answer: Option
_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

Manager
Manager
avatar
Joined: 24 May 2013
Posts: 79
GMAT ToolKit User
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 17 Mar 2016, 00:00
If x is a positive integer, what is the units digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)?

((24^2)*(33)*(17)*(9^2))^x * (24*33*17^2)
Considering only unit digits
(6*3*7*1)^x * (4*3*9)

Again reducing to unit digits
6^x * 8
8

Hence D.

Thanks
Director
Director
User avatar
G
Joined: 23 Jan 2013
Posts: 598
Schools: Cambridge'16
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 18 Mar 2016, 02:17
(24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)=?

2x+1=odd, so 4^odd=4 as unit

2x=even, so 9^even=1 as unit

x+1 and x+2 means that exponent of 7 is one more than exponent of 3. If we look cyclicity we always get 7 in unit when multiplying

So, 4*1*7=8 as unit

D
Intern
Intern
avatar
Joined: 06 Mar 2015
Posts: 27
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 29 May 2016, 08:27
kevincan wrote:
Remember that in the GMAT, it pays to look for shortcuts!

33^(x+1)*17^(x+2)= ((33*17)^(x+1))*17 which has a units digit of 7!


I don't understand the shortcut. Could you kindly enunciate?
Manager
Manager
User avatar
Joined: 18 May 2016
Posts: 67
Concentration: Finance, International Business
GMAT 1: 720 Q49 V39
GPA: 3.7
WE: Analyst (Investment Banking)
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 31 May 2016, 13:41
nishi999 wrote:
kevincan wrote:
Remember that in the GMAT, it pays to look for shortcuts!

33^(x+1)*17^(x+2)= ((33*17)^(x+1))*17 which has a units digit of 7!


I don't understand the shortcut. Could you kindly enunciate?


Basically, 17^(x+2) = 17^(x+1) * 17 (we take one 17 away from the power to get x+1 instead of x+2

Thus, 33^(x+1) x 17^(x+2) = 33^(x+1) x 17^(x+1) * 17 = (33*17)^(x+1) * 17

We can do the same with (24)^(2x + 1) * (9)^(2x) = (24*9)^(2x) * 24

From (33*17)^(x+1) * 17 we take first two unit digits, first, 3 x 7 = 21, then 1 * 7 = 7
From (24*9)^(2x) * 24 we take first two unit digits, first, 4 * 9 = 36, then 6 * 4 = 24
Finally, we have 7 and 24 or 7 * 4 = 28

Answer: 8

I hope I could help :)
_________________

Please kindly +Kudos if my posts or questions help you!

My debrief: Self-study: How to improve from 620(Q39,V36) to 720(Q49,V39) in 25 days!

Intern
Intern
avatar
B
Joined: 28 May 2017
Posts: 6
GMAT 1: 640 Q44 V34
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 28 Aug 2017, 05:30
Hi
There is a relatively easier way.
As x is a positive number, take x=1.
So, 24^3 * 33^2 * 17^3 * 19^2 = 4*9*3*1 = 8.
Cheers :)
Target Test Prep Representative
User avatar
G
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2781
Re: If x is a positive integer, what is the units digit of  [#permalink]

Show Tags

New post 31 Aug 2017, 10:19
MA wrote:
If x is a positive integer, what is the units digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)?

A. 4
B. 6
C. 7
D. 8
E. 9


Since we are only concerned with the units digit, we can simplify the expression as:

(4)^(2x + 1)*(3)^(x + 1)*(7)^(x + 2)*(9)^(2x)

This simplified expression will have the same units digit as the given expression. Next, let’s look at the units digit patterns of powers of 4, 3, 7, and 9, respectively:

Units digits of powers of 4: 4-6 (the patterns repeats in a cycle of 2 with 4^odd = 4 and 4^even = 6)

Units digits of powers of 3: 3-9-7-1 (the patterns repeats in a cycle of 4 with 3^(a multiple of 4) = 1)

Units digits of powers of 7: 7-9-3-1 (the patterns repeats in a cycle of 4 with 7^(a multiple of 4) = 1)

Units digits of powers of 9: 9-1 (the patterns repeats in a cycle of 2 with 9^odd = 9 and 9^even = 1)

Since 2x + 1 is always odd regardless of what integer x is, 4^(2x + 1) = 4^odd = 4. Similarly, since 2x is always even regardless of what integer x is, 9^(2x) = 9^even = 1. However, since x + 1 (the exponent of 3) and x + 2 (the exponent of 7) are sometimes odd and sometimes even depending on what integer x is, we are going to change tactics in analyzing the units digit of (3)^(x + 1)*(7)^(x + 2). Notice that:

(3)^(x + 1)*(7)^(x + 2) = (3)^(x + 1)*(7)^(x + 1)*7 = (3*7)^(x + 1)*(7) = (21)^(x + 1)*(7)

Since we are only concerned with the units digit, we can simplify (21)^(x + 1)*(7) as (1)^(x + 1)*(7). Since 1 raised to any power is 1, the units digit of (1)^(x + 1)*(7) or (21)^(x + 1)*(7) is 1*7 = 7. With this, we can see that the units digit of (4)^(2x + 1)*[(3)^(x + 1)*(7)^(x + 2)]*(9)^(2x) is 4*[7]*1 = 28, i.e., 8.

Answer: D
_________________

Jeffery Miller
Head of GMAT Instruction

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Re: If x is a positive integer, what is the units digit of &nbs [#permalink] 31 Aug 2017, 10:19

Go to page    1   2    Next  [ 22 posts ] 

Display posts from previous: Sort by

If x is a positive integer, what is the units digit of

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  

Events & Promotions

PREV
NEXT


GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.