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# What is the tens digit of 6^17?

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17 Oct 2010, 11:29
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What is the tens digit of 6^17?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9
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17 Oct 2010, 15:32
Didn't find a shorter way:
6^17= (6*6*6*6*6)^5*6=7286^3*6
We need only the tenth digit therefore- 86*86*86*6= 7396*86
96*86=...56--- 56*6= ...36
Correct answer is B - 6

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17 Oct 2010, 15:36
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Raths wrote:
what is the ten's digit of 6^17

1. 1
2. 3
3. 5
4. 7
5. 9

Just like cyclicity of the last digit, we can observe the cyclicity of the last 2 digits in this case : (which is possible because there is no cyclicity in the unit's digit, it is always 6)
6^2 = 36
6^3 = 16
6^4 = 96
6^5 = 76
6^6 = 56
6^7 = 36
.. and then it repeats

So for 6^17, it will have the same tens digit as 6^12, 6^7, 6^2 ... or 3

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17 Oct 2010, 20:41
shrouded can you please explain it a bit more, didnt get why 6^17, it will have the same tens digit as 6^12, 6^7, 6^2 .. thank you in advance

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18 Oct 2010, 00:42
So the logic here is simple. Consider the number 6^x, lets say that you know the tens digit of this number, can you find out the tens digit of 6^(x+1) ?

What we know is that the last digit of 6^x will always be 6 (which is easy enough to see). Now the fact of the matter is that the ten's digit of 6^(x+1) is only dependent on the tens digit of 6^x.

Because Ten's digit of 6^(x+1) = 6*(Ten's digit of 6^x) + 3 (carried over during the multiplication of the units digits 6 with the new 6).

Like 6^3 = 216
So 6^4, units digit is 6 and ten's digit is 6*1+3 = 9

Hence, as soon as the ten's digit of 6^x becomes the same as the ten's digit of 6^y, the pattern of tens digit will start to repeat itself

6^2 = 36
6^3 = 16
6^4 = 96
6^5 = 76
6^6 = 56
6^7 = 36

The pattern here is 3,1,9,7,5,3,1,9,7,5,3,1,9,7,5,3,1,9,7,5,3.....
The cyclicity of the pattern is five, so every 5th element in this series will be the same hence 2nd,7th,12th,17th have to be the same
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What is the tens digit of 6^17? [#permalink]

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03 Feb 2012, 20:44
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What is the tens digit of 6^17?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

Last edited by Bunuel on 09 Mar 2014, 11:55, edited 2 times in total.

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04 Feb 2012, 05:30
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Smita04 wrote:
What is the tens digit of 6^17?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition.

The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}:
The tens digit of 6^2=36 is 3;
The tens digit of 6^3=216 is 1;
The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits);
The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit);
The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits);
The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits).

Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3.

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Re: What is the tens digit of 6^17? [#permalink]

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10 Feb 2012, 03:14
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Bunuel,can you calculate it with modulo as below:

1) periodicity of the ten's digit is 5
2) 17 mod 5 = 2
3) 6^17 will have the same digit as 6^2

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Re: What is the tens digit of 6^17? [#permalink]

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10 Feb 2012, 05:10
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well, this question demands calculation to see a pattern of tens digits
keep calculating till it's confirmed that u have hit a pattern.
6^1 = 6
6^2 = 36
6^3 = 216
now don't multiply 216 by 6, rather we are interested in only first two digits to know the outcome so
6^4 = 96 ( 16 x 6)
6^5 = 576 ( 96 x 6)
6^ 6 = 456 ( 76 x 6)
7^ 6 =336 ( 56 x 6)
so now we have the pattern in tens digit i.e.
3 in (6^2),
1 in (6^3),
9 in (6^4),
7 in (6^5),
5 in (6^6),
3 in (6^7),

so the tens digit is 3 for the 2,7,12 and 17 times..
IMO B
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Re: What is the tens digit of 6^17? [#permalink]

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10 Feb 2012, 07:16
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nonameee wrote:
Bunuel,can you calculate it with modulo as below:

1) periodicity of the ten's digit is 5
2) 17 mod 5 = 2
3) 6^17 will have the same digit as 6^2

Yes, that's correct.
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Re: What is the tens digit of 6^17? [#permalink]

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21 Dec 2012, 07:38
Smita04 wrote:
What is the tens digit of 6^17?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

$$6^2 = 36$$
$$6^3 = 36 * 6 = n16$$
$$6^5 = 6^2 * 6^3 = 36 * n16 = n76$$

Note that when you multiply, you don't have to finish it all the way, knowing the tens digit should suffice....
Also, using the table we have we can calculate $$6^{10}$$ and $$6^{17}$$. We work with what we already have above/

$$6^10 = 6^5 * 6^5 = n76 * n76 = n76$$
$$6^7 = 6^5 * 6^2 = n36 * n76 = n36$$
$$6^{17} = 6^{10} * 6^{7} = n76 * n36$$ (We already know what happens to n76 * n36 as calculated above...) $$=n36$$

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Last edited by mbaiseasy on 13 Jan 2013, 23:11, edited 4 times in total.

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Re: What is the tens digit of 6^17? [#permalink]

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13 Jan 2013, 13:15
bunuel would you please post me a link on the topic of exponents and powers from gmat math book if it has been finished..i want to learn and master way u hav solved the problem

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Re: What is the tens digit of 6^17? [#permalink]

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14 Jan 2013, 01:44
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chiccufrazer1 wrote:
bunuel would you please post me a link on the topic of exponents and powers from gmat math book if it has been finished..i want to learn and master way u hav solved the problem

Posted from my mobile device

For more on number theory and exponents check: http://gmatclub.com/forum/math-number-theory-88376.html

DS questions on exponents: http://gmatclub.com/forum/search.php?se ... &tag_id=39
PS questions on exponents: http://gmatclub.com/forum/search.php?se ... &tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: http://gmatclub.com/forum/tough-and-tri ... 25967.html
Tough and tricky PS exponents and roots questions with detailed solutions: http://gmatclub.com/forum/tough-and-tri ... 25956.html

Hope it helps.
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17 Jan 2013, 10:59
Bunuel wrote:
Smita04 wrote:
What is the tens digit of 6^17?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition.

The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}:
The tens digit of 6^2=36 is 3;
The tens digit of 6^3=216 is 1;
The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits);
The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit);
The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits);
The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits).

Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3.

i have noticed that every number has 6 as the unit digit..is it the same for other numbers that they repeat each of the unit's digit throughout when it is being raised to powers of consecutive integers

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18 Jan 2013, 04:28
chiccufrazer1 wrote:
Bunuel wrote:
Smita04 wrote:
What is the tens digit of 6^17?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition.

The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}:
The tens digit of 6^2=36 is 3;
The tens digit of 6^3=216 is 1;
The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits);
The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit);
The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits);
The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits).

Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3.

i have noticed that every number has 6 as the unit digit..is it the same for other numbers that they repeat each of the unit's digit throughout when it is being raised to powers of consecutive integers

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No. You could test that very easily yourself. Is the units digit of 2^2 equal 2? No, its 4.

• 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.

For more check here: http://gmatclub.com/forum/math-number-theory-88376.html

Hope it helps.
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Re: What is the tens digit of 6^17? [#permalink]

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09 Mar 2014, 13:08
Bumping for review and further discussion.

For more on this kind of questions check Units digits, exponents, remainders problems collection.
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Re: What is the tens digit of 6^17? [#permalink]

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02 Jul 2015, 10:56
Bunuel wrote:
Smita04 wrote:
What is the tens digit of 6^17?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition.

The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}:
The tens digit of 6^2=36 is 3;
The tens digit of 6^3=216 is 1;
The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits);
The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit);
The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits);
The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits).

Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3.

.

Hi Bunuel,

Can we do this in following way

6^17 = (2*3)^17

(2^16*3^16) *2*3

now 2 repeats in pattern 2,4,8,6 and 3 repeats in pattern 3,9,7,1 so when we multiply 2^16*3^16 last digit is 6 now multiply 6 *6 so its 36 so tens digit is 3.

Thanks

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Re: What is the tens digit of 6^17? [#permalink]

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20 Mar 2016, 02:05
Smita04 wrote:
What is the tens digit of 6^17?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

6^4 when divided by 100 the remainder is -4
So we have to check -4^4*6 when divided by 100
or 256*6 = 1536

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Re: What is the tens digit of 6^17? [#permalink]

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21 Apr 2017, 02:22
6^17 = 16,926,659,444,736 & its tenth digit is 3.

option b is correct.

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Re: What is the tens digit of 6^17? [#permalink]

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13 Sep 2017, 23:46
If Question like this appears in GMAT. and we are asked to find the last two digits this could be used

Rule Express Even numbers in the form ( 2^10) ^even which will have last two digits as 76 or ( 2^10) ^odd where last two digits is 24

For Odd numbers Express them 3^4k, 7 ^ 4k, 9 ^2k

Question was 6^17

So (2^ 17 ) ( 3^17)

= {(2^10)^1 * 2^7} { (3^4)^3 * 3^5}

So last two digits (2^10)^1= 24
So last two digits 2^7= 28

Last two digits of this number (3^4)^3 = (81)^3= last two digits are 41 (1 will be the last digit and second last digit will be 4 that i got by multiplying 8 *3
last two digits of this number 3^5 = 43

So (24 * 28) * (41* 43)

Don't do complete multiplication just do it till you get two digits

72 * 63= 36

So digit in tenth place is 36
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Re: What is the tens digit of 6^17?   [#permalink] 13 Sep 2017, 23:46

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