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What is the tens digit of 36^10? [#permalink]
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02 Feb 2015, 07:25
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Re: What is the tens digit of 36^10? [#permalink]
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02 Feb 2015, 10:17
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
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 "calculationbased" 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 Final Answer: While this type of approach is not particularly elegant, if you're comfortable multiplying 2digt numbers together, the work isn't that bad. It's also preferable to staring at the screen for 12 minutes and then blindly guessing. GMAT assassins aren't born, they're made, Rich
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Re: What is the tens digit of 36^10? [#permalink]
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02 Feb 2015, 13:13
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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



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Re: What is the tens digit of 36^10? [#permalink]
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02 Feb 2015, 14:35
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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



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Re: What is the tens digit of 36^10? [#permalink]
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03 Feb 2015, 00:20
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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
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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
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Re: What is the tens digit of 36^10? [#permalink]
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04 Feb 2015, 03:42
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we go like this: 36^10=36^(2*5) 36^2=1296, since we need only the tens digit we take 96 96100 leeds us to 4; the value of 4^5 is the same as 2^10, which leeds us to 1024. 10024=76. we need the tenths digit, so we take 7. D



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Re: What is the tens digit of 36^10? [#permalink]
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09 Feb 2015, 04:34
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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 patternrecognition, "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
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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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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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What is the tens digit of 36^10? [#permalink]
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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



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Re: What is the tens digit of 36^10? [#permalink]
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12 Aug 2016, 03:09
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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, MoMNote 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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\(36^{10}\) = \(6^{20}\)=\((2*3)^{20}\) 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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