What is the tens digit of 6^17? : GMAT Problem Solving (PS)
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# What is the tens digit of 6^17?

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

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03 Feb 2012, 19:44
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What is the tens digit of 6^17?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Mar 2014, 10:55, edited 2 times in total.
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04 Feb 2012, 04:30
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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, 02:14
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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, 04:10
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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, 04:23
I know how to find the cyclicity of the ten's digit. I just wanted to know whether steps 2) and 3) can be used:

Quote:
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, 05:15
1. Identify pattern (first one does not have a tens, 3, 1, 9, 7, 5, 3)
2. scale up to 6^17 = tens digit 3

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

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10 Feb 2012, 06:16
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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, 06:38
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, 22:11, 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, 12:15
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, 00:44
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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: math-number-theory-88376.html

DS questions on exponents: search.php?search_id=tag&tag_id=39
PS questions on exponents: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Hope it helps.
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17 Jan 2013, 09:59
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, 03:28
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

Posted from my mobile device

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: 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, 12:08
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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04 May 2015, 19:22
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Re: What is the tens digit of 6^17? [#permalink]

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02 Jul 2015, 09:56
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, 01:05
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
Re: What is the tens digit of 6^17?   [#permalink] 20 Mar 2016, 01:05
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