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# What is the tens digit of 11^{13}?

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Math Expert
Joined: 02 Sep 2009
Posts: 49252
What is the tens digit of 11^{13}?  [#permalink]

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02 Sep 2015, 22:26
3
6
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Difficulty:

35% (medium)

Question Stats:

72% (00:45) correct 28% (00:43) wrong based on 264 sessions

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What is the tens digit of $$11^{13}$$?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Kudos for a correct solution.

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Joined: 07 Apr 2015
Posts: 175
What is the tens digit of 11^{13}?  [#permalink]

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Updated on: 03 Sep 2015, 02:41
1
In this case ciclicity is helpful.

11^1 = 11
11^2 = 121
11^3 = 1331
11^4 = 14641

As one can recognize then, the tens digit is equal to the exponents digit. 11^11 is equal to 11^1 tens digit wise, therefore 11^13 has the same tens digit as 11^3, which would be 3.

Originally posted by noTh1ng on 03 Sep 2015, 01:48.
Last edited by noTh1ng on 03 Sep 2015, 02:41, edited 1 time in total.
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What is the tens digit of 11^{13}?  [#permalink]

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03 Sep 2015, 02:34
2
$$11^2$$ has 2 in Tens digit
$$11^3$$ has 3 in Tens digit
$$11^9$$ has 9 in Tens digit
11^11 has 1 in Tens digit
Since the Tens digit of the number is same as the units digit of the power of 11, Therefore, 11^13 must have 3 in Tens digit.

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Re: What is the tens digit of 11^{13}?  [#permalink]

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03 Sep 2015, 09:37
2
1
Bunuel wrote:
What is the tens digit of $$11^{13}$$?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Kudos for a correct solution.

$$11^1$$ = 11
$$11^2$$ = 121
$$11^3$$ = 1331

When the power of 11 is increased by 1, the 10's digit will increase by 1.
This will continue till $$11^9$$.
For $$11^{10}$$, we will get 0 at ten's place.
Same cycle will again continue for next powers of 11.
So,
10's digit of $$11^{13}$$ will be same as 10's digit of $$11^3$$ i.e 3

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Re: What is the tens digit of 11^{13}?  [#permalink]

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03 Sep 2015, 11:19
1
Solution:

11^n will have units digit of n as its tens digit.
Ex. 11^2 = 121 , 11^3 = 1331... so on.
11^13 will have 3 as tens digit.

Option C
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Joined: 12 Nov 2013
Posts: 40
Re: What is the tens digit of 11^{13}?  [#permalink]

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05 Sep 2015, 11:02
1
Bunuel wrote:
What is the tens digit of $$11^{13}$$?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Kudos for a correct solution.

11 ^13 can be written as 11^8 x 11^4 x 11^1

Now, last two digits of 11 raise to power are

11 = 11
11^2 = 21
11 ^4 = 21 * 21 = 41
11 ^ 8 = 41 * 41 = 81

SO 11 ^13 = 81 * 41 * 11 = last two digits = 31

So the tens digit is 3.

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Math Expert
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Posts: 49252
Re: What is the tens digit of 11^{13}?  [#permalink]

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07 Sep 2015, 04:01
Bunuel wrote:
What is the tens digit of $$11^{13}$$?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

This question may well test the upper level of GMAT difficulty, but can be solved using the large-number-with-exponents strategy of finding patterns. Look at the progression of the easier-to-calculate powers of 11:

$$11^{1}= 11$$

$$11^{2}= 121$$

$$11^{3} = 1331$$

$$11^{4}= 14641$$

As you’re looking at digits places, you should first note that the units digit of each is 1, which will hold true anytime you’re multiplying a units digit of 1 by a units digit of 1. Now look at the tens places. They maintain the value of the exponent and progress from 1 to 2 to 3 to 4. They simply increase by one each time. Now, you may well just assume that the pattern will continue to to hold, and after this many in a row with your knowledge that exponents are very pattern-driven, you’re likely to be correct. But for the sake of thoroughness, let’s look at why that rule holds. Each time you multiply a number by 11, you’re multiplying it by 10+1. Therefore, you can multiply it by 1, multiply it by 10, and add those products together. Multiplying by 1 is easy (it just stays the same) and multiplying by 10 is the same as multiplying by 1 and just adding a 0 at the end. Therefore, $$11^{3}$$, or $$11^{2}*11$$, is:

121*1 = 121

121*10 = 1210

121 + 1210 = 1331

Watching the tens place, you should notice that the *10 portion just takes the previous units digit (which is always 1) and shifts it to the tens place, and that the *1 portion keeps the previous tens place. Essentially, we use the *10 portion to add 1 to the tens digit each time, ensuring that the pattern will hold, and the tens digit of any power of 11 will simply be the units digit of the exponent.

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Re: What is the tens digit of 11^{13}?  [#permalink]

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16 Oct 2015, 12:18
Hi Bunuel,

Thank you for the in detail explanation.

About this - "tens digit of any power of 11 will simply be the units digit of the exponent"

do we have any other numbers with such interesting facts ?

Thanks,
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Posts: 3481
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Schools: Chicago (Booth) - Class of 2011
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Re: What is the tens digit of 11^{13}?  [#permalink]

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15 Sep 2018, 10:00
11^13 = (10 + 1)^13 = 10^13+...+130+1

It's C (3)

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Re: What is the tens digit of 11^{13}? &nbs [#permalink] 15 Sep 2018, 10:00
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