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

 It is currently 19 Oct 2019, 08:44

### 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

# What is the unit's digit of 7^75 + 6 ?

Author Message
TAGS:

### Hide Tags

Intern
Joined: 22 Jun 2010
Posts: 37
What is the unit's digit of 7^75 + 6 ?  [#permalink]

### Show Tags

Updated on: 03 Jul 2013, 00:52
15
00:00

Difficulty:

5% (low)

Question Stats:

77% (00:53) correct 23% (01:02) wrong based on 759 sessions

### HideShow timer Statistics

What is the unit's digit of $$7^{75} + 6$$ ?

A. 1
B. 3
C. 5
D. 7
E. 9

(C) 2008 GMAT Club - m12#29

I put the official explanation and the part I do not understand (blue text) in a spoiler

$$7^1$$ ends with 7

$$7^2$$ ends with 9

$$7^3$$ ends with 3

$$7^4$$ ends with 1

$$7^5$$ ends with 7

...

$$7^{76}$$ ends with 1. --> ???

So, $$7^{75}$$ ends with 3. --> ???

$$7^{75} + 6$$ ends with 9.

Originally posted by AndreG on 14 Sep 2010, 12:58.
Last edited by Bunuel on 03 Jul 2013, 00:52, edited 1 time in total.
Renamed the topic and edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 58445
Re: GMAT CLUB TEST m12#29 - last digit  [#permalink]

### Show Tags

14 Sep 2010, 13:42
4
7
sandeep800 wrote:
AndreG wrote:
What is the unit's digit of $$7^{75} + 6$$ ?

(C) 2008 GMAT Club - m12#29

* 1
* 3
* 5
* 7
* 9

I put the official explanation and the part I do not understand (blue text) in a spoiler

$$7^1$$ ends with 7

$$7^2$$ ends with 9

$$7^3$$ ends with 3

$$7^4$$ ends with 1

$$7^5$$ ends with 7

...

$$7^{76}$$ ends with 1. --> ???

So, $$7^{75}$$ ends with 3. --> ???

$$7^{75} + 6$$ ends with 9.

well i will say that whatever may be the number if we have to find the last digit of some number whose power isgiven..then the best method is to divide the power by 4 since all the digits from 1...9 will surely repeat after every 4th digit...
then raise the digit to the power of remainder...
here 75/4 remainder=3
7^3=last digit comes out to be 3
now 3+6=9

thanx

The above is correct with a little correction: when remainder is zero, then we should rise to the power not of remainder 0 but to the power of the cyclicity number.

For example las digit of 7^24 is the same as the last digit of 7^4 as the cyclicity of 7 in power is 4 and 24 divided by 4 gives remainder of zero.

From Number Theory chapter of Math Book:

LAST DIGIT OF A POWER

Determining the last digit of $$(xyz)^n$$:

1. Last digit of $$(xyz)^n$$ is the same as that of $$z^n$$;
2. Determine the cyclicity number $$c$$ of $$z$$;
3. Find the remainder $$r$$ when $$n$$ divided by the cyclisity;
4. When $$r>0$$, then last digit of $$(xyz)^n$$ is the same as that of $$z^r$$ and when $$r=0$$, then last digit of $$(xyz)^n$$ is the same as that of $$z^c$$, where $$c$$ is the cyclisity number.

• Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base.
• Integers ending with 2, 3, 7 and 8 have a cyclicity of 4.
• Integers ending with 4 (eg. $$(xy4)^n$$) have a cyclisity of 2. When n is odd $$(xy4)^n$$ will end with 4 and when n is even $$(xy4)^n$$ will end with 6.
• Integers ending with 9 (eg. $$(xy9)^n$$) have a cyclisity of 2. When n is odd $$(xy9)^n$$ will end with 9 and when n is even $$(xy9)^n$$ will end with 1.

Example: What is the last digit of $$127^{39}$$?
Solution: Last digit of $$127^{39}$$ is the same as that of $$7^{39}$$. Now we should determine the cyclisity of $$7$$:

1. 7^1=7 (last digit is 7)
2. 7^2=9 (last digit is 9)
3. 7^3=3 (last digit is 3)
4. 7^4=1 (last digit is 1)
5. 7^5=7 (last digit is 7 again!)
...

So, the cyclisity of 7 is 4.

Now divide 39 (power) by 4 (cyclisity), remainder is 3.So, the last digit of $$127^{39}$$ is the same as that of the last digit of $$7^{39}$$, is the same as that of the last digit of $$7^3$$, which is $$3$$.

Hope it helps.
_________________
##### General Discussion
Math Expert
Joined: 02 Sep 2009
Posts: 58445
Re: GMAT CLUB TEST m12#29 - last digit  [#permalink]

### Show Tags

14 Sep 2010, 13:20
3
AndreG wrote:
What is the unit's digit of $$7^{75} + 6$$ ?

(C) 2008 GMAT Club - m12#29

* 1
* 3
* 5
* 7
* 9

I put the official explanation and the part I do not understand (blue text) in a spoiler

$$7^1$$ ends with 7

$$7^2$$ ends with 9

$$7^3$$ ends with 3

$$7^4$$ ends with 1

$$7^5$$ ends with 7

...

$$7^{76}$$ ends with 1. --> ???

So, $$7^{75}$$ ends with 3. --> ???

$$7^{75} + 6$$ ends with 9.

7 in power repeats pattern of 4: 7-9-3-1. As 75=4*18+3 then the last digit of $$7^{75}$$ is the same as the last digit of $$7^3$$, which is 3. Units digit of $$7^{75} + 6$$ will be: 3 plus 6 = 9.

For more on this issue check Number Theory chapter of Math Book (link in my signature).

Hope it helps.
_________________
Senior Manager
Status: GMAT Time...!!!
Joined: 03 Apr 2010
Posts: 264
Schools: Chicago,Tuck,Oxford,cambridge
Re: GMAT CLUB TEST m12#29 - last digit  [#permalink]

### Show Tags

14 Sep 2010, 13:26
AndreG wrote:
What is the unit's digit of $$7^{75} + 6$$ ?

(C) 2008 GMAT Club - m12#29

* 1
* 3
* 5
* 7
* 9

I put the official explanation and the part I do not understand (blue text) in a spoiler

$$7^1$$ ends with 7

$$7^2$$ ends with 9

$$7^3$$ ends with 3

$$7^4$$ ends with 1

$$7^5$$ ends with 7

...

$$7^{76}$$ ends with 1. --> ???

So, $$7^{75}$$ ends with 3. --> ???

$$7^{75} + 6$$ ends with 9.

well i will say that whatever may be the number if we have to find the last digit of some number whose power isgiven..then the best method is to divide the power by 4 since all the digits from 1...9 will surely repeat after every 4th digit...
then raise the digit to the power of remainder...
here 75/4 remainder=3
7^3=last digit comes out to be 3
now 3+6=9

thanx
Senior Manager
Status: GMAT Time...!!!
Joined: 03 Apr 2010
Posts: 264
Schools: Chicago,Tuck,Oxford,cambridge
Re: GMAT CLUB TEST m12#29 - last digit  [#permalink]

### Show Tags

14 Sep 2010, 13:55
1
Bunuel wrote:
sandeep800 wrote:
AndreG wrote:
What is the unit's digit of $$7^{75} + 6$$ ?

(C) 2008 GMAT Club - m12#29

* 1
* 3
* 5
* 7
* 9

I put the official explanation and the part I do not understand (blue text) in a spoiler

$$7^1$$ ends with 7

$$7^2$$ ends with 9

$$7^3$$ ends with 3

$$7^4$$ ends with 1

$$7^5$$ ends with 7

...

$$7^{76}$$ ends with 1. --> ???

So, $$7^{75}$$ ends with 3. --> ???

$$7^{75} + 6$$ ends with 9.

well i will say that whatever may be the number if we have to find the last digit of some number whose power isgiven..then the best method is to divide the power by 4 since all the digits from 1...9 will surely repeat after every 4th digit...
then raise the digit to the power of remainder...
here 75/4 remainder=3
7^3=last digit comes out to be 3
now 3+6=9

thanx

The above is correct with a little correction: when remainder is zero, then we should rise to the power not of remainder 0 but to the power of the cyclicity number.

For example las digit of 7^24 is the same as the last digit of 7^4 as the cyclicity of 7 in power is 4 and 24 divided by 4 gives remainder of zero.

From Number Theory chapter of Math Book:

LAST DIGIT OF A POWER

Determining the last digit of $$(xyz)^n$$:

1. Last digit of $$(xyz)^n$$ is the same as that of $$z^n$$;
2. Determine the cyclicity number $$c$$ of $$z$$;
3. Find the remainder $$r$$ when $$n$$ divided by the cyclisity;
4. When $$r>0$$, then last digit of $$(xyz)^n$$ is the same as that of $$z^r$$ and when $$r=0$$, then last digit of $$(xyz)^n$$ is the same as that of $$z^c$$, where $$c$$ is the cyclisity number.

• Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base.
• Integers ending with 2, 3, 7 and 8 have a cyclicity of 4.
• Integers ending with 4 (eg. $$(xy4)^n$$) have a cyclisity of 2. When n is odd $$(xy4)^n$$ will end with 4 and when n is even $$(xy4)^n$$ will end with 6.
• Integers ending with 9 (eg. $$(xy9)^n$$) have a cyclisity of 2. When n is odd $$(xy9)^n$$ will end with 9 and when n is even $$(xy9)^n$$ will end with 1.

Example: What is the last digit of $$127^{39}$$?
Solution: Last digit of $$127^{39}$$ is the same as that of $$7^{39}$$. Now we should determine the cyclisity of $$7$$:

1. 7^1=7 (last digit is 7)
2. 7^2=9 (last digit is 9)
3. 7^3=3 (last digit is 3)
4. 7^4=1 (last digit is 1)
5. 7^5=7 (last digit is 7 again!)
...

So, the cyclisity of 7 is 4.

Now divide 39 (power) by 4 (cyclisity), remainder is 3.So, the last digit of $$127^{39}$$ is the same as that of the last digit of $$7^{39}$$, is the same as that of the last digit of $$7^3$$, which is $$3$$.

Hope it helps.

Thanx a lot bunuel for correcting me..i wud have applied my method in GMAT if u had not corrected me....
Intern
Joined: 22 Jun 2010
Posts: 37
Re: GMAT CLUB TEST m12#29 - last digit  [#permalink]

### Show Tags

14 Sep 2010, 14:06
Wow, you guys helped me a lot! THANKS!!
Manager
Joined: 20 Jul 2010
Posts: 191
Re: GMAT CLUB TEST m12#29 - last digit  [#permalink]

### Show Tags

14 Sep 2010, 15:23
Thanks for summarising the concept. I used to calculate what you call cyclicity in every problem and reach my conclusions
_________________
If you like my post, consider giving me some KUDOS !!!!! Like you I need them
Math Expert
Joined: 02 Sep 2009
Posts: 58445
What is the unit's digit of 7^75 + 6 ?  [#permalink]

### Show Tags

09 Mar 2014, 13:09
For more on this kind of questions check Units digits, exponents, remainders problems collection.
_________________
Board of Directors
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4773
Location: India
GPA: 3.5
Re: What is the unit's digit of 7^75 + 6 ?  [#permalink]

### Show Tags

23 Nov 2016, 11:45
AndreG wrote:
What is the unit's digit of $$7^{75} + 6$$ ?

A. 1
B. 3
C. 5
D. 7
E. 9

Since , the cyclicity of 7 is 4

The units digit of $$7^{75} = 3$$

So, Units digit will be 3+ 6 = 9

Hence, answer will be (E) 9...

_________________
Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )
Current Student
Joined: 12 Aug 2015
Posts: 2567
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Re: What is the unit's digit of 7^75 + 6 ?  [#permalink]

### Show Tags

23 Jan 2017, 18:22
Nice Question.
Here is what i did in this one ->
Cyclicity of 7 is 4 =>
7
9
3
1
Hence the units digit of 7^75 => 7^4m+3 will be 3.
So 7^75+6 will have 3+6=9 as its units digit.

Hence E.

_________________
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15281
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: What is the unit's digit of 7^75 + 6 ?  [#permalink]

### Show Tags

17 Mar 2018, 11:38
Hi All,

The GMAT doesn't expect you to calculate the value of 7^75; instead, let's try to figure out the pattern behind this math…. The question asks for the UNITS DIGIT, so we'll focus on that…

7^1 = 7
7^2 = 49
7^3 = 343
7^4 = ends in a 1

7^5 = ends in a 7
7^6 = ends in a 9
etc.

The pattern is 7931 7931 7931 etc.

So the units digit will follow this repeating pattern of 4 units digits…

75/4 = 18r3

Thus, the 75th number will be the "third number" in the pattern…which is a 3

3+6 = 9

GMAT assassins aren't born, they're made,
Rich
_________________
Contact Rich at: Rich.C@empowergmat.com

The Course Used By GMAT Club Moderators To Earn 750+

souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★
Retired Moderator
Joined: 27 Oct 2017
Posts: 1259
Location: India
GPA: 3.64
WE: Business Development (Energy and Utilities)
What is the unit's digit of 7^75 + 6 ?  [#permalink]

### Show Tags

17 Mar 2018, 11:50
Hi
To solve these types of questions, it is not necessary to remember cyclicity order of different digits,
In fact, all digits follow a common cyclicity order, they repeat itself after 4k+1 power.
Steps to solve such questions:
1) divide the power by 4 and find remainder.
here the remainder is 3
2) Now find the unit digit by raising it to exponent of remainder (if remainder is 0, raise it to exponent 4)
here, it is 7^3 = 3

this method works for every digit

(PS: for some of digits , we have simpler pattern method.
1) 0 - always 0
2) 4 - odd power = 4, even power = 6
3) 5 always 5
4) 6 always 6
5) 9 - odd power 9, even power = 1)
_________________
Non-Human User
Joined: 09 Sep 2013
Posts: 13278
Re: What is the unit's digit of 7^75 + 6 ?  [#permalink]

### Show Tags

08 May 2019, 05:15
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: What is the unit's digit of 7^75 + 6 ?   [#permalink] 08 May 2019, 05:15
Display posts from previous: Sort by