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# What is the tens digit of 36^10?

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Math Expert
Joined: 02 Sep 2009
Posts: 58415
What is the tens digit of 36^10?  [#permalink]

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02 Feb 2015, 07:25
6
18
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Difficulty:

95% (hard)

Question Stats:

45% (02:13) correct 55% (02:07) wrong based on 342 sessions

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What is the tens digit of 36^10?

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

Kudos for a correct solution.

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Re: What is the tens digit of 36^10?  [#permalink]

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02 Feb 2015, 12:51
7
4
Hi All,

The GMAT does not expect you to do an excessive amount of calculation to find the answer to a question, although you might be asked to do a reasonable amount of arithmetic in certain cases). When you're faced with this type of "calculation-based" question, if you can't find the hidden pattern or the "elegant" approach, you still have to opportunity to just do math. Be sure to note that there's a point at which you can STOP doing math though.....as long as you pay attention to the question that is ASKED.

Here, we're asked for the TENS DIGIT of 36^10.

We're clearly NOT going to calculate this entire product, but we can do some math and take advantage of how math "works"....

36^10 can be rewritten as....(36^4)(36^4)(36^2)

36^2 is a product that you should be able to calculate....

(36)(36) = 1296

Since the question is asking about the TENS DIGIT, anything to the "left" of the TENS DIGIT really doesn't matter, so we can SKIP THOSE DIGITS...

(36)(36) = (......96) a number that ends in 96.

36^4 = (36^2)(36^2)

We know that 36^2 ends in .....96, so we're really just multiplying....
(...96)(...96)

AND we're going to ignore every digit EXCEPT for the TENS and UNITS digits....(try doing the math; remember to stop working once you have those 2 digits figured out)....

(...96)(...96) = (......16)

Now we know that....
36^2 ends in ....96
36^4 ends in ....16

So we have a little more "math" to go....

(36^4)(36^4)(36^2) = (...16)(...16)(...96)

(...16)(...16) = ...56

(...56)(...96) = ...76

So, the TENS DIGIT = 7

While this type of approach is not particularly elegant, if you're comfortable multiplying 2-digt numbers together, the work isn't that bad. It's also preferable to staring at the screen for 1-2 minutes and then blindly guessing.

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Re: What is the tens digit of 36^10?  [#permalink]

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02 Feb 2015, 14:35
7
6
Bunuel wrote:
What is the tens digit of 36^10?

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

Kudos for a correct solution.

36^10 = 6^20

(6^2)=6*6 = 36
(6^3)= 36*6 = .16
(6^4)= .16*6 = ..96
(6^5) = ..96*6 = ..76
(6^6) = ..76*6 = ...56
(6^7) = ....56*6 = ....36

If you see there is a pattern here in tens digits 3,1,9,7,5,3,1 and so on...

Continue the pattern up to 6^20 ( dont actually calculate full values) and answer is D: 7

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Re: What is the tens digit of 36^10?  [#permalink]

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02 Feb 2015, 10:17
2
1
hi the ans is D.. and the solution can be..
basically what question asks us is to find the remainder when divided by 100..
36^10= 96^5(as the remainder when 36^2 is divided by 100)=(-4)^5= -24 or 76..
so the tens digit is 7
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Re: What is the tens digit of 36^10?  [#permalink]

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02 Feb 2015, 13:13
1
IMO its D
although I did performing a lengthy calculation which took some time.. hence waiting for a crisp approach and also a short method to find 10ens digit for any three digit or even 7 digit number.
Thanks
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Re: What is the tens digit of 36^10?  [#permalink]

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03 Feb 2015, 00:20
1
Bunuel wrote:
What is the tens digit of 36^10?

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

Kudos for a correct solution.

36^10 = 6^20
If you type powers of six they end in
6
36
16
96
76
56
the pattern is 3-->1-->9-->7-->5 so for 36 we start with 3-->9-->5-->1-->7 and repeat.
36^10 will come at 7. Answer D.
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Re: What is the tens digit of 36^10?  [#permalink]

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03 Feb 2015, 23:54
2

$$36^{10} = 6^{20}$$

$$6^1 = 06$$

$$6^2 = 36$$

$$6^3 = ..16$$

$$6^4 = ...96$$

$$6^5 = ...76$$

$$6^6 = ...56$$

$$6^7 = ...36$$

By following the pattern above, tens digit would be 7
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Re: What is the tens digit of 36^10?  [#permalink]

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04 Feb 2015, 03:42
2
we go like this:
36^10=36^(2*5)
36^2=1296, since we need only the tens digit we take 96
96-100 leeds us to -4; the value of -4^5 is the same as 2^10, which leeds us to 1024. 100-24=76. we need the tenths digit, so we take 7.
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Re: What is the tens digit of 36^10?  [#permalink]

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09 Feb 2015, 04:34
1
4
Bunuel wrote:
What is the tens digit of 36^10?

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

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

Since this problem asks for a specific digit that isn't a units digit, you should see this as a pattern-recognition, "create your own number property" problem. And one instinct should tell you that you can simplify 36^10 by rephrasing it as (6^2)^10, or 6^20. This gives you smaller numbers to work with as you establish a pattern.

From there, find the pattern with a keen eye on the tens digit:

6^2=36
6^3=216
6^4=...96 (Just multiply the last two digits since we only care about the tens digit)
6^5=...76
6^6=...56
6^7=...36
and hence starts the cycle again:

3, 1, 9, 7, 5,

3, 1, 9, 7, 5,

and so on

Since the pattern starts with the 6^2 (6^1 has no tens digit) and the 20th power isn't that much to jot down, you may want to simply write out that pattern until you get to the 20th number:

0, 3, 1, 9, 7, 5, 3, 1, 9, 7, 5, 3, 1, 9, 7, 5, 3, 1, 9, 7

The tens digit, then, will be 7.
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Re: What is the tens digit of 36^10?  [#permalink]

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09 Feb 2015, 05:59
2
4
Celestial09 wrote:
IMO its D
although I did performing a lengthy calculation which took some time.. hence waiting for a crisp approach and also a short method to find 10ens digit for any three digit or even 7 digit number.
Thanks
Celestial

hi..
whereever u are asked for last digit / tens digit / hundreds digit, it means remainder when div by 10 / 100 / 1000....
basically what question asks us is to find the remainder when divided by 100.. and we should be concerned only with last two digits..

36^10=(36^2)^5= 96^5(as the remainder when 36^2 is divided by 100)
=(-4)^5(-4 is the remainder when 96 div by 100)
= -24 or 76..
so the tens digit is 7
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Re: What is the tens digit of 36^10?  [#permalink]

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09 Aug 2016, 11:37
chetan2u wrote:
Celestial09 wrote:
IMO its D
although I did performing a lengthy calculation which took some time.. hence waiting for a crisp approach and also a short method to find 10ens digit for any three digit or even 7 digit number.
Thanks
Celestial

hi..
whereever u are asked for last digit / tens digit / hundreds digit, it means remainder when div by 10 / 100 / 1000....
basically what question asks us is to find the remainder when divided by 100.. and we should be concerned only with last two digits..

36^10=(36^2)^5= 96^5(as the remainder when 36^2 is divided by 100)
=(-4)^5(-4 is the remainder when 96 div by 100)
= -24 or 76..
so the tens digit is 7

Hello Chetan,

In the above steps, can you please clarify how did you reached out to -24 or 76?

36^10=(36^2)^5= 96^5 (96 is the remainder when 36^2 is divided by 100)
=(-4)^5(-4 is the remainder when 96 div by 100)
= -24 or 76..???
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Posts: 7975
Re: What is the tens digit of 36^10?  [#permalink]

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09 Aug 2016, 18:54
Manonamission wrote:
chetan2u wrote:
Celestial09 wrote:
IMO its D
although I did performing a lengthy calculation which took some time.. hence waiting for a crisp approach and also a short method to find 10ens digit for any three digit or even 7 digit number.
Thanks
Celestial

hi..
whereever u are asked for last digit / tens digit / hundreds digit, it means remainder when div by 10 / 100 / 1000....
basically what question asks us is to find the remainder when divided by 100.. and we should be concerned only with last two digits..

36^10=(36^2)^5= 96^5(as the remainder when 36^2 is divided by 100)
=(-4)^5(-4 is the remainder when 96 div by 100)
= -24 or 76..
so the tens digit is 7

Hello Chetan,

In the above steps, can you please clarify how did you reached out to -24 or 76?

36^10=(36^2)^5= 96^5 (96 is the remainder when 36^2 is divided by 100)
=(-4)^5(-4 is the remainder when 96 div by 100)
= -24 or 76..???

Hi,
When we multiply -4*-4=16..
And 16*16=...56
Nowfifth -4 is left..
When 56 is multiplied by -4, we get -24 as last two digits..
But remainder cannot be NEGATIVE, so add to 100..
100+(-24)=76
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Re: What is the tens digit of 36^10?  [#permalink]

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09 Aug 2016, 22:49
Check out my 3 part blog post discussing various ways of finding the last two digits:

http://www.veritasprep.com/blog/2014/12 ... ns-part-i/
http://www.veritasprep.com/blog/2014/12 ... s-part-ii/
http://www.veritasprep.com/blog/2014/12 ... -part-iii/
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What is the tens digit of 36^10?  [#permalink]

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10 Aug 2016, 11:59
VeritasPrepKarishma wrote:

Hello Karishma,

I was able to get through powers of 11 note in the link provided. However, I have some queries related to
Last two digits of 6^58
Clearly the cyclicity of ten's digit is 5

3 1 9 7 5 3 1 9 7 5

"The new cycle with tens digit of 3 begins at the powers of 2, 7, 12, 17, 22, 27 etc. So the new cycle will also begin at power of 57 and 6^58 will have 1 as the tens digit."

Can you elaborate this more? How the new cycle for 6^58 will start with at a power of 57?
( Sorry for asking this but somehow I am unable to grasp this concept )

regards,
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Re: What is the tens digit of 36^10?  [#permalink]

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12 Aug 2016, 03:09
1
Manonamission wrote:
VeritasPrepKarishma wrote:

Hello Karishma,

I was able to get through powers of 11 note in the link provided. However, I have some queries related to
Last two digits of 6^58
Clearly the cyclicity of ten's digit is 5

3 1 9 7 5 3 1 9 7 5

"The new cycle with tens digit of 3 begins at the powers of 2, 7, 12, 17, 22, 27 etc. So the new cycle will also begin at power of 57 and 6^58 will have 1 as the tens digit."

Can you elaborate this more? How the new cycle for 6^58 will start with at a power of 57?
( Sorry for asking this but somehow I am unable to grasp this concept )

regards,
MoM

Note that 6^1 has no tens digit.
The cyclicity is 3, 1, 9, 7, 5 but it starts from 6^2.
6^2 has tens digit of 3.
6^7 has tens digit of 3.
6^12 has tens digit of 3.
...

So the new cycle starts at 2, 7, 12, 17, 22, 27, 32, 37.... Note the symmetry (after every 5 powers).
The new cycle will start at 52 and then at 57 too.
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What is the tens digit of 36^10?  [#permalink]

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01 Nov 2016, 02:16
1
$$36^{10}$$ = $$6^{20}$$=$$(2*3)^{20}$$
Now we have 2 separate cases
$$2^{20}$$=$$(2^{10})^2$$=$$1024^2$$ 24 raise to even power will always end in 76.
$$3^{20}$$=$$(3^4)^5$$=$$81^5$$ Units digit 1 raised to any power will give 1. Tens digit is equal to the units digit of the product of 8 and 5.
So we have 76*01=76
Tens digit is 7.
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Re: What is the tens digit of 36^10?  [#permalink]

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08 Apr 2019, 04:15
Bunuel wrote:
What is the tens digit of 36^10?

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

Kudos for a correct solution.

36^10 = 4^10*9^10 = = 2^20*3^20 = 76*01 = 76

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Re: What is the tens digit of 36^10?   [#permalink] 08 Apr 2019, 04:15
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