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# What is the tens digit of 36^10?

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Math Expert
Joined: 02 Sep 2009
Posts: 46064
What is the tens digit of 36^10? [#permalink]

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02 Feb 2015, 07:25
6
10
00:00

Difficulty:

95% (hard)

Question Stats:

47% (01:31) correct 53% (01:26) wrong based on 387 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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Math Expert
Joined: 02 Aug 2009
Posts: 5875
Re: What is the tens digit of 36^10? [#permalink]

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02 Feb 2015, 10:17
1
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, 12:51
7
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.

GMAT assassins aren't born, they're made,
Rich
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# Rich Cohen

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** Retired Moderator Status: I Declare War!!! Joined: 02 Apr 2014 Posts: 242 Location: United States Concentration: Finance, Economics GMAT Date: 03-18-2015 WE: Asset Management (Investment Banking) Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 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 Celestial Manager Joined: 24 Dec 2014 Posts: 129 Location: India Concentration: International Business, Finance GMAT 1: 700 Q47 V39 GMAT 2: 790 Q51 V50 GPA: 3.65 WE: Operations (Energy and Utilities) Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 02 Feb 2015, 14:35 7 3 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 Posted from my mobile device Current Student Joined: 02 Sep 2014 Posts: 89 Location: United States Schools: Haas EWMBA '20 GMAT 1: 770 Q50 V44 GPA: 3.97 Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 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. SVP Status: The Best Or Nothing Joined: 27 Dec 2012 Posts: 1837 Location: India Concentration: General Management, Technology WE: Information Technology (Computer Software) Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 03 Feb 2015, 23:54 2 Answer = D = 7 $$36^{10} = 6^{20}$$ $$6^1 = 06$$ $$6^2 = 36$$ $$6^3 = ..16$$ $$6^4 = ...96$$ $$6^5 = ...76$$ $$6^6 = ...56$$ $$6^7 = ...36$$ By following the pattern above, tens digit would be 7 _________________ Kindly press "+1 Kudos" to appreciate Manager Joined: 17 Dec 2013 Posts: 58 GMAT Date: 01-08-2015 Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 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. D Math Expert Joined: 02 Sep 2009 Posts: 46064 Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 09 Feb 2015, 04:34 1 2 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. _________________ Math Expert Joined: 02 Aug 2009 Posts: 5875 Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 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 _________________ Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372 Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html GMAT online Tutor Manager Joined: 11 Jul 2016 Posts: 80 Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 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..??? Math Expert Joined: 02 Aug 2009 Posts: 5875 Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 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 _________________ Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372 Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html GMAT online Tutor Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8100 Location: Pune, India Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 09 Aug 2016, 22:49 Manager Joined: 11 Jul 2016 Posts: 80 What is the tens digit of 36^10? [#permalink] ### Show Tags 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, MoM Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8100 Location: Pune, India Re: What is the tens digit of 36^10? [#permalink] ### Show Tags 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. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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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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11 Mar 2018, 08:44
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Re: What is the tens digit of 36^10?   [#permalink] 11 Mar 2018, 08:44
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