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# The tens digit of 6^17 is

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Math Expert
Joined: 02 Sep 2009
Posts: 54493
The tens digit of 6^17 is  [#permalink]

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17 Nov 2014, 12:42
00:00

Difficulty:

75% (hard)

Question Stats:

60% (02:05) correct 40% (02:03) wrong based on 134 sessions

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Tough and Tricky questions: Exponents/Powers.

The tens digit of 6^17 is

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

Kudos for a correct solution.

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Re: The tens digit of 6^17 is  [#permalink]

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17 Nov 2014, 22:40
3
2
Bunuel wrote:

Tough and Tricky questions: Exponents/Powers.

The tens digit of 6^17 is

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

Kudos for a correct solution.

For tens digit, pattern recognition is very effective.

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

The new cycle with tens digit of 3 begins at the powers of 2, 7, 12, 17 etc
So 6^17 will have 3 as the tens digit.

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Re: The tens digit of 6^17 is  [#permalink]

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17 Nov 2014, 23:06
1
Bunuel wrote:

Tough and Tricky questions: Exponents/Powers.

The tens digit of 6^17 is

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

Kudos for a correct solution.

I got B as well. Started with 6, 36, 216, 1296, ....76, ...56, ...36. The pattern is 3, 1, 9, 7, 5, ...... So 6^2 has same tens digit as 6^17 which is 3.

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Re: The tens digit of 6^17 is  [#permalink]

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18 Nov 2014, 01:18
2

$$6^1 = 6$$

$$6^2$$ = 36

$$6^3$$ = 216

$$6^4$$>> Units place = 16*6 = 9

$$6^5$$ >> Units place = 96*6 = 7

$$6^6$$ >> Units place = 76*6 = 5

$$6^7$$ >> Units place = 56*6 = 3

Cyclicity is 3, 1, 9, 7, 5

$$6^2, 6^7, 6^{12}, 6^{17}$$ will have the same digit in units place = 3

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Re: The tens digit of 6^17 is  [#permalink]

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18 Nov 2014, 01:42
1
In order to find tense place, we can check cyclicity of tense place of 6^1, 6^2, 6^3 etc.

Let us find out.

6^2 = 36 --> tense place is 3
6^3 = 216 --> tense place is 1
6^4= ..296 --> tense place is 9
6^5= ....76 --> tense place is 7
6^6=.....56 --> tense place is 5
6^7=.....36 --> tense place is 3 (Please note, here last two digits are 36, which is also same for 6^2)
6^8=.....16 --> tense place is 1 ( same as in 6^3)

so we can see that last two digits are repeating after 6^7
i.e. last two digit of 6^2= last two digit of 6^7
last two digit of 6^3= last two digit of 6^8
last two digit of 6^4= last two digit of 6^9
so on.

So cyclicity of tense place is 5.

So 6^2, 6^7, 6^12, 6^17 will have 3 in tense place.

Regards,
Ammu
Math Expert
Joined: 02 Sep 2009
Posts: 54493
Re: The tens digit of 6^17 is  [#permalink]

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18 Nov 2014, 08:02
Official Solution:

The tens digit of $$6^{17}$$ is

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

We know that there must be a pattern, since we can’t be expected to expand $$6^{17}$$ out to all its digits. In other words, we must be able to spot a repeating cycle of digits.

The only way forward is to compute tens digits for powers of 6, starting with $$6^1$$, and see what we get. To go up, multiply the previous result by 6 and drop any higher digits than the tens, but we have to keep the units digit (which, as we’ll see, will be 6 every time).

$$6^1 = 6$$ (no tens digit)

$$6^2 = 6 \times 6^1 = 36$$ (tens digit = 3)

$$6^3 = 6 \times 6^2 = ..16$$ (tens digit = 1)

$$6^4 = 6 \times 6^3 = ..96$$ (tens digit = 9)

$$6^5 = 6 \times 6^4 = ..76$$ (tens digit = 7)

$$6^6 = 6 \times 6^5 = ..56$$ (tens digit = 5)

$$6^7 = 6 \times 6^6 = ..36$$ (tens digit = 3)

Whew - the numbers finally started repeating! The cycle is 3, 1, 9, 7, 5 - which is 5 terms long. Every power will have the same tens digit as the 5th larger power, so $$6^2$$, $$6^7$$, $$6^{12}$$, and most importantly $$6^{17}$$ will all have 3 as their tens digit.

Notice that the pattern didn’t start until $$6^2$$. $$6^1$$ doesn’t have a tens digit (or has a tens digit of 0, but this digit is never repeated later in the cycle).

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Re: The tens digit of 6^17 is  [#permalink]

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24 Jun 2017, 08:48
I think multiplying till finding a pattern would not be a sure way to solve.
the method i used is congruency

6: 6mod100
6^2: 36mod100
6^4: 96mod100
6^8: 16mod100
6^16: 56mod100
6^17: 36mod100

so 36 would be the last two digits of the final number.
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Re: The tens digit of 6^17 is  [#permalink]

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21 Mar 2019, 07:03
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Re: The tens digit of 6^17 is   [#permalink] 21 Mar 2019, 07:03
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