It is currently 20 Nov 2017, 14:34

### 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 last digit 3^{3^3} ? A. 1 B. 3 C. 6 D. 7 E. 9

Author Message
Manager
Joined: 12 Feb 2008
Posts: 177

Kudos [?]: 47 [4], given: 0

What is the last digit 3^{3^3} ? A. 1 B. 3 C. 6 D. 7 E. 9  [#permalink]

### Show Tags

23 Sep 2008, 07:17
4
KUDOS
13
This post was
BOOKMARKED
What is the last digit $$3^{3^3}$$ ?

A. 1
B. 3
C. 6
D. 7
E. 9

OA is D (7).

Kudos [?]: 47 [4], given: 0

Current Student
Joined: 28 Dec 2004
Posts: 3345

Kudos [?]: 322 [1], given: 2

Location: New York City
Schools: Wharton'11 HBS'12
Re: last digit of a power [#permalink]

### Show Tags

23 Sep 2008, 07:23
1
KUDOS
3^27

so lets see 27mod4=3..

3^1=3
3^2=9
3^3=7

Unit digit is 7

Kudos [?]: 322 [1], given: 2

SVP
Joined: 07 Nov 2007
Posts: 1790

Kudos [?]: 1088 [1], given: 5

Location: New York
Re: last digit of a power [#permalink]

### Show Tags

23 Sep 2008, 07:29
1
KUDOS
elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?

* 1
* 3
* 6
* 7
* 9

27^3 ....--> 7*7*7 --> unit digit xx3 ..

B
_________________

Smiling wins more friends than frowning

Kudos [?]: 1088 [1], given: 5

VP
Joined: 30 Jun 2008
Posts: 1031

Kudos [?]: 729 [7], given: 1

Re: last digit of a power [#permalink]

### Show Tags

23 Sep 2008, 07:32
7
KUDOS
elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?

* 1
* 3
* 6
* 7
* 9

3^3 = 27

the last digit of $$3^{3^3}$$ will be the last digit of (27)^3.

now the last digit of 27*27*27 will be the same as last digit of 7*7*7 = 343 (ie) 3

Another example we can use here to understand this concept better (I made this example up)

What is the last digit of 39*87*81?

A. 2
B. 3
C. 4
D. 5
E. 6

To find the last digit of 39*87*81. All we have to do is, multiply the last digits of 39, 87 and 81

when we multiply, 9*7 we get 63. Now multiply the last digits of 63 and 81, i.e. 3*1 we get 3

so the last digit of 39*87*81 will be 3.
_________________

"You have to find it. No one else can find it for you." - Bjorn Borg

Kudos [?]: 729 [7], given: 1

Intern
Joined: 20 Sep 2008
Posts: 6

Kudos [?]: 6 [1], given: 0

Re: last digit of a power [#permalink]

### Show Tags

23 Sep 2008, 12:20
1
KUDOS
is the above mentioned method right ?

thanks

Kudos [?]: 6 [1], given: 0

Senior Manager
Joined: 31 Jul 2008
Posts: 290

Kudos [?]: 57 [1], given: 0

Re: last digit of a power [#permalink]

### Show Tags

23 Sep 2008, 14:59
1
KUDOS
i dnt think that we can reduce the above statement into 27^3 becoz that is (3^3)^3 which is not what the question says...........

so the answer has to be 7

Kudos [?]: 57 [1], given: 0

VP
Joined: 30 Jun 2008
Posts: 1031

Kudos [?]: 729 [2], given: 1

Re: last digit of a power [#permalink]

### Show Tags

23 Sep 2008, 21:10
2
KUDOS
elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?

$$3^{3^3}$$ should be taken as 3^27 ?

in that case divide 27 by 4. The remainder is 3. Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7.

So 7 is the right answer.
_________________

"You have to find it. No one else can find it for you." - Bjorn Borg

Kudos [?]: 729 [2], given: 1

Manager
Joined: 12 Feb 2008
Posts: 177

Kudos [?]: 47 [1], given: 0

Re: last digit of a power [#permalink]

### Show Tags

24 Sep 2008, 05:24
1
KUDOS
amitdgr wrote:
elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?

$$3^{3^3}$$ should be taken as 3^27 ?

in that case divide 27 by 4. The remainder is 3. Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7.

So 7 is the right answer.

why" Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7."

where do you get this from

Kudos [?]: 47 [1], given: 0

SVP
Joined: 07 Nov 2007
Posts: 1790

Kudos [?]: 1088 [4], given: 5

Location: New York
Re: last digit of a power [#permalink]

### Show Tags

24 Sep 2008, 06:22
4
KUDOS
sorry guys. Its my mistake.

$${3}^{3^3}$$ can't be taken as $$27^3..$$It should be $$3^{27}$$
$$3^9 = 27^3$$

3^1=3
3^2=9
3^3=27
3^4=81.
..
..
3^27 = ....7
_________________

Smiling wins more friends than frowning

Kudos [?]: 1088 [4], given: 5

VP
Joined: 30 Jun 2008
Posts: 1031

Kudos [?]: 729 [12], given: 1

Re: last digit of a power [#permalink]

### Show Tags

24 Sep 2008, 06:41
12
KUDOS
elmagnifico wrote:
amitdgr wrote:
elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?

$$3^{3^3}$$ should be taken as 3^27 ?

in that case divide 27 by 4. The remainder is 3. Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7.

So 7 is the right answer.

why" Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7."

where do you get this from

$$3^1$$ =3,
$$3^2$$ =9,
$$3^3$$ =27,
$$3^4$$ =81,
$$3^5$$ =243,
$$3^6$$ =729, and so on .....

Now it is not humanly possible to remember all the numbers till $$3^27$$

If you have noticed in the above series the last digit repeats after every 4 terms

the last digit is same for $$3^5$$ and $$3^1$$
the last digit is same for $$3^6$$ and $$3^2$$

If 3 is the unit digit of a number then the unit digit repeats every fourth consecutive term.For our convenience here, lets call it cyclicity. So 3 has a cyclicity of 4.

To find the unit digit of a number having 3 as its last digit and raised to a positive power, divide the power by 4 and find the remainder.

If the remainder is 1 then the unit digit is same as of the unit digit of $$3^1$$
If the remainder is 2 then the unit digit is same as of the unit digit of $$3^2$$ and so on .....

Note that if the remainder is "0" then the unit digit is same as $$3^4$$ since the cyclicity is 4.

Also remember that the numbers 2,3,7 and 8 have cyclicity of 4

in our problem above we have 3^27

3 has a cyclicity of 4 so divide the number 27 by 4. We get a remainder of 3. Now as per cyclicity the last digit of 3^27 is same as that of 3^3. 3^3 is 27 so the last digit of 3^27 is 7.
_________________

"You have to find it. No one else can find it for you." - Bjorn Borg

Last edited by amitdgr on 24 Sep 2008, 06:58, edited 1 time in total.

Kudos [?]: 729 [12], given: 1

VP
Joined: 30 Jun 2008
Posts: 1031

Kudos [?]: 729 [43], given: 1

Re: last digit of a power [#permalink]

### Show Tags

24 Sep 2008, 06:50
43
KUDOS
4
This post was
BOOKMARKED
Lets take another example

Find the last digit of 122^94

A. 2
B. 4
C. 6
D. 8
E. 9

Now the last digit of 122 is 2. We require only this number to determine the last digit of 122 raised to a positive power.

so the problem is essentially reduced to find the last digit of 2^94.

Now we know 2 has a cyclicity of 4. So we divide 94 by 4. The remainder for 94/4 is 2.

so last digit of 2^94 is same as that of 2^2 which is 4.

so last digit of 122^94 is 4

Remember:
1) Numbers 2,3,7 and 8 have a cyclicity of 4
2) Numbers 0,1,5 and 6 have a cyclicity of 1 (ie) all the powers will have the same unit digit. eg. 5^245 will have "5" as unit digit, 5^2000 will aslo have "5" as unit digit. Same holds for 0,1 and 6
3) If 4 is the number in the unit place of the base number then the unit digit will be "4" if the power is odd and it will be "6" if the power is even. eg. 4^123 will have unit digit of 4 since 123 is odd.
4) Similarly, for 9 the unit digit will be "9" for odd powers and "1" for even powers. eg 9^234 has unit digit as "1" since 234 is even.
_________________

"You have to find it. No one else can find it for you." - Bjorn Borg

Kudos [?]: 729 [43], given: 1

Intern
Joined: 20 Sep 2008
Posts: 6

Kudos [?]: 6 [4], given: 0

Re: last digit of a power [#permalink]

### Show Tags

24 Sep 2008, 07:04
4
KUDOS
amitdgr wrote:
Lets take another example

Find the last digit of 122^94

A. 2
B. 4
C. 6
D. 8
E. 9

Now the last digit of 122 is 2. We require only this number to determine the last digit of 122 raised to a positive power.

so the problem is essentially reduced to find the last digit of 2^94.

Now we know 2 has a cyclicity of 4. So we divide 94 by 4. The remainder for 94/4 is 2.

so last digit of 2^94 is same as that of 2^2 which is 4.

so last digit of 122^94 is 4

Remember:
1) Numbers 2,3,7 and 8 have a cyclicity of 4
2) Numbers 0,1,5 and 6 have a cyclicity of 1 (ie) all the powers will have the same unit digit. eg. 5^245 will have "5" as unit digit, 5^2000 will aslo have "5" as unit digit. Same holds for 0,1 and 6
3) If 4 is the number in the unit place of the base number then the unit digit will be "4" if the power is odd and it will be "6" if the power is even. eg. 4^123 will have unit digit of 4 since 123 is odd.
4) Similarly, for 9 the unit digit will be "9" for odd powers and "1" for even powers. eg 9^234 has unit digit as "1" since 234 is even.

Wow !! Awesome I tried out this thing with a few numbers and matched the results with my scientific calculator. This method gives perfect answers.

You deserve at least a dozen KUDOS for typing out all this patiently and sharing this knowledge with all of us.

+1 from me. Guys pour in Kudos for this

Chayanika

Kudos [?]: 6 [4], given: 0

Intern
Joined: 14 Sep 2007
Posts: 42

Kudos [?]: 5 [1], given: 0

Re: last digit of a power [#permalink]

### Show Tags

24 Sep 2008, 07:56
1
KUDOS
its 3

(3^3)^3 = 19683

whats the OA!?

Kudos [?]: 5 [1], given: 0

Manager
Joined: 12 Feb 2008
Posts: 177

Kudos [?]: 47 [4], given: 0

Re: last digit of a power [#permalink]

### Show Tags

24 Sep 2008, 08:25
4
KUDOS
1
This post was
BOOKMARKED
chayanika wrote:
amitdgr wrote:
Lets take another example

Find the last digit of 122^94

A. 2
B. 4
C. 6
D. 8
E. 9

Now the last digit of 122 is 2. We require only this number to determine the last digit of 122 raised to a positive power.

so the problem is essentially reduced to find the last digit of 2^94.

Now we know 2 has a cyclicity of 4. So we divide 94 by 4. The remainder for 94/4 is 2.

so last digit of 2^94 is same as that of 2^2 which is 4.

so last digit of 122^94 is 4

Remember:
1) Numbers 2,3,7 and 8 have a cyclicity of 4
2) Numbers 0,1,5 and 6 have a cyclicity of 1 (ie) all the powers will have the same unit digit. eg. 5^245 will have "5" as unit digit, 5^2000 will aslo have "5" as unit digit. Same holds for 0,1 and 6
3) If 4 is the number in the unit place of the base number then the unit digit will be "4" if the power is odd and it will be "6" if the power is even. eg. 4^123 will have unit digit of 4 since 123 is odd.
4) Similarly, for 9 the unit digit will be "9" for odd powers and "1" for even powers. eg 9^234 has unit digit as "1" since 234 is even.

Wow !! Awesome I tried out this thing with a few numbers and matched the results with my scientific calculator. This method gives perfect answers.

You deserve at least a dozen KUDOS for typing out all this patiently and sharing this knowledge with all of us.

+1 from me. Guys pour in Kudos for this

Chayanika

i agree with you. he deserves many kudos.

thanks a million.

the OA is 7 indeed. what a wonderful explanation.

Kudos [?]: 47 [4], given: 0

Manager
Joined: 12 Feb 2008
Posts: 177

Kudos [?]: 47 [4], given: 0

Re: last digit of a power [#permalink]

### Show Tags

24 Sep 2008, 08:27
4
KUDOS
elmagnifico wrote:
amitdgr wrote:
elmagnifico wrote:
What is the last digit $$3^{3^3}$$ ?

$$3^{3^3}$$ should be taken as 3^27 ?

in that case divide 27 by 4. The remainder is 3. Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7.

So 7 is the right answer.

why" Now the last digit of 3^27 is same as that of 3^3 and that happens to be 7."

where do you get this from

$$3^1$$ =3,
$$3^2$$ =9,
$$3^3$$ =27,
$$3^4$$ =81,
$$3^5$$ =243,
$$3^6$$ =729, and so on .....

Now it is not humanly possible to remember all the numbers till $$3^27$$

If you have noticed in the above series the last digit repeats after every 4 terms

the last digit is same for $$3^5$$ and $$3^1$$
the last digit is same for $$3^6$$ and $$3^2$$

If 3 is the unit digit of a number then the unit digit repeats every fourth consecutive term.For our convenience here, lets call it cyclicity. So 3 has a cyclicity of 4.

To find the unit digit of a number having 3 as its last digit and raised to a positive power, divide the power by 4 and find the remainder.

If the remainder is 1 then the unit digit is same as of the unit digit of $$3^1$$
If the remainder is 2 then the unit digit is same as of the unit digit of $$3^2$$ and so on .....

Note that if the remainder is "0" then the unit digit is same as $$3^4$$ since the cyclicity is 4.

Also remember that the numbers 2,3,7 and 8 have cyclicity of 4

in our problem above we have 3^27

3 has a cyclicity of 4 so divide the number 27 by 4. We get a remainder of 3. Now as per cyclicity the last digit of 3^27 is same as that of 3^3. 3^3 is 27 so the last digit of 3^27 is 7.[/quote]

by the way, i have never had such a GMAT rush. it is like a sugar rush.
EVERY ONE GIVE MORE KUDOS HERE

Kudos [?]: 47 [4], given: 0

Intern
Joined: 20 Sep 2008
Posts: 1

Kudos [?]: 1 [1], given: 0

Re: last digit of a power [#permalink]

### Show Tags

24 Sep 2008, 09:06
1
KUDOS
Very neat method amitdgr +2 from me

Kudos [?]: 1 [1], given: 0

Manager
Joined: 30 Jun 2009
Posts: 51

Kudos [?]: 21 [1], given: 6

Re: last digit of a power [#permalink]

### Show Tags

26 Jul 2009, 21:14
1
KUDOS
Quote:
Remember:
1) Numbers 2,3,7 and 8 have a cyclicity of 4
2) Numbers 0,1,5 and 6 have a cyclicity of 1 (ie) all the powers will have the same unit digit. eg. 5^245 will have "5" as unit digit, 5^2000 will aslo have "5" as unit digit. Same holds for 0,1 and 6
3) If 4 is the number in the unit place of the base number then the unit digit will be "4" if the power is odd and it will be "6" if the power is even. eg. 4^123 will have unit digit of 4 since 123 is odd.
4) Similarly, for 9 the unit digit will be "9" for odd powers and "1" for even powers. eg 9^234 has unit digit as "1" since 234 is even.

So glad I come across this thread! Great tip! Thanks a bunch +1

Kudos [?]: 21 [1], given: 6

Intern
Joined: 02 Aug 2009
Posts: 8

Kudos [?]: 23 [5], given: 5

Re: last digit of a power [#permalink]

### Show Tags

05 Aug 2009, 15:11
5
KUDOS
1
This post was
BOOKMARKED
Here's another way of looking at it !

Here the given number is $$(xyz)^n$$
z is the last digit of the base.
n is the index

To find out the last digit in $$(xyz)^n$$, the following steps are to be followed.
Divide the index (n) by 4, then

Case I
If remainder = 0
then check if z is odd (except 5), then last digit = 1
and if z is even then last digit = 6

Case II
If remainder = 1, then required last digit = last digit of the base (i.e. z)
If remainder = 2, then required last digit = last digit of the base $$(z)^2$$
If remainder = 3, then required last digit = last digit of the base $$(z)^3$$

Note : If z = 5, then the last digit in the product = 5

Example:
Find the last digit in (295073)^130

Solution: Dividing 130 by 4, the remainder = 2
Refering to Case II, the required last digit is the last digit of $$(z)^2$$, ie $$(3)^2$$ = 9 , (because z = 3)

Kudos [?]: 23 [5], given: 5

Manager
Joined: 18 Jul 2009
Posts: 167

Kudos [?]: 116 [2], given: 37

Location: India
Schools: South Asian B-schools
Re: last digit of a power [#permalink]

### Show Tags

05 Aug 2009, 20:25
2
KUDOS
3^3^3 to solve this we have to take top down approach...we cannot deduce 27^3......it should be 3^27....hence answer is 7
_________________

Bhushan S.
If you like my post....Consider it for Kudos

Kudos [?]: 116 [2], given: 37

Senior Manager
Joined: 11 Dec 2008
Posts: 476

Kudos [?]: 256 [1], given: 12

Location: United States
GMAT 1: 760 Q49 V44
GPA: 3.9
Re: last digit of a power [#permalink]

### Show Tags

05 Aug 2009, 21:43
1
KUDOS
Is 3^3^3 definitely taken as 3^(3^3)?

Is reading it as (3^3)^3 incorrect?

Kudos [?]: 256 [1], given: 12

Re: last digit of a power   [#permalink] 05 Aug 2009, 21:43

Go to page    1   2   3   4    Next  [ 70 posts ]

Display posts from previous: Sort by

# What is the last digit 3^{3^3} ? A. 1 B. 3 C. 6 D. 7 E. 9

Moderator: Bunuel

 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®.