Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 26 May 2017, 10:20

### 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

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

Author Message
TAGS:

### Hide Tags

Manager
Joined: 22 Jun 2010
Posts: 57
Followers: 1

Kudos [?]: 57 [0], given: 10

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

### Show Tags

14 Sep 2010, 12:58
8
This post was
BOOKMARKED
00:00

Difficulty:

15% (low)

Question Stats:

65% (01:37) correct 35% (00:48) wrong based on 639 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

[Reveal] 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.
[Reveal] Spoiler: OA

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: 38908
Followers: 7739

Kudos [?]: 106216 [3] , given: 11613

Re: GMAT CLUB TEST m12#29 - last digit [#permalink]

### Show Tags

14 Sep 2010, 13:20
3
KUDOS
Expert's post
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

[Reveal] 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: 293
Schools: Chicago,Tuck,Oxford,cambridge
Followers: 6

Kudos [?]: 53 [0], given: 7

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

[Reveal] 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
Math Expert
Joined: 02 Sep 2009
Posts: 38908
Followers: 7739

Kudos [?]: 106216 [0], given: 11613

Re: GMAT CLUB TEST m12#29 - last digit [#permalink]

### Show Tags

14 Sep 2010, 13:42
Expert's post
4
This post was
BOOKMARKED
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

[Reveal] 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.
_________________
Senior Manager
Status: GMAT Time...!!!
Joined: 03 Apr 2010
Posts: 293
Schools: Chicago,Tuck,Oxford,cambridge
Followers: 6

Kudos [?]: 53 [0], given: 7

Re: GMAT CLUB TEST m12#29 - last digit [#permalink]

### Show Tags

14 Sep 2010, 13:55
1
This post was
BOOKMARKED
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

[Reveal] 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....
Manager
Joined: 22 Jun 2010
Posts: 57
Followers: 1

Kudos [?]: 57 [0], given: 10

Re: GMAT CLUB TEST m12#29 - last digit [#permalink]

### Show Tags

14 Sep 2010, 14:06
Wow, you guys helped me a lot! THANKS!!
Senior Manager
Joined: 20 Jul 2010
Posts: 264
Followers: 2

Kudos [?]: 87 [0], given: 9

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: 38908
Followers: 7739

Kudos [?]: 106216 [0], given: 11613

Re: What is the unit's digit of 7^75 + 6 ? [#permalink]

### Show Tags

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

For more on this kind of questions check Units digits, exponents, remainders problems collection.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15463
Followers: 649

Kudos [?]: 209 [0], given: 0

Re: What is the unit's digit of 7^75 + 6 ? [#permalink]

### Show Tags

02 Apr 2015, 18:38
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: 15463
Followers: 649

Kudos [?]: 209 [0], given: 0

Re: What is the unit's digit of 7^75 + 6 ? [#permalink]

### Show Tags

06 Jun 2016, 18:59
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.
_________________
Math Forum Moderator
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 2678
Location: India
GPA: 3.5
Followers: 111

Kudos [?]: 860 [0], given: 324

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 )

BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 2136
Followers: 73

Kudos [?]: 611 [0], given: 553

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.

_________________

Give me a hell yeah ...!!!!!

Re: What is the unit's digit of 7^75 + 6 ?   [#permalink] 23 Jan 2017, 18:22
Similar topics Replies Last post
Similar
Topics:
2 If the units digit of x^3 is 6, what is the units digit of integer x? 3 23 Jan 2017, 18:13
8 What is the units digit of (5!*4! + 6!*5!)/31? 7 23 Jan 2017, 19:49
What is the units digit of (23^6)(17^3)(61^9)? 2 07 Mar 2016, 09:08
7 The integer x has a tens digit of 6 and a units digit of 7. The units 5 25 Nov 2016, 20:36
15 What is the units digit of 6^15 - 7^4 - 9^3? 19 19 Sep 2016, 17:40
Display posts from previous: Sort by