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# If n is a positive integer, what is the remainder when 3^n is divided

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Math Expert
Joined: 02 Sep 2009
Posts: 46251
If n is a positive integer, what is the remainder when 3^n is divided [#permalink]

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21 Aug 2017, 06:15
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15% (low)

Question Stats:

79% (00:58) correct 21% (00:35) wrong based on 66 sessions

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If n is a positive integer, what is the remainder when 3^n is divided by 10?

(1) n is a multiple of 8
(2) n is a multiple of 12

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If n is a positive integer, what is the remainder when 3^n is divided [#permalink]

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21 Aug 2017, 07:42
3^1(3) divided by 10 gives us a remainder of 3
3^2(9) divided by 10 gives us a remainder of 9
3^3(27) divided by 10 gives us a remainder of 7
3^4(81) divided by 10 gives us a remainder of 1
3^5(243) divided by 10 gives us a remainder of 3
3^6(729) divided by 10 gives us a remainder of 9
If you look there is a pattern of remainders for the different numbers 3,9,7,1,3,9.....

1. n is a multiple of 8
n can be 8,16,24.... and so on
The remainder will be the same when 3^n is divided by 10 if the values of n are evenly spaced.
Hence, this statement alone is sufficient

2. n is a multiple of 12
n can be 12,24,36.... and so on
The remainder will be the same when 3^n is divided by 10 if the values of n are evenly spaced.
Hence, this statement alone is sufficient(Option D)
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If n is a positive integer, what is the remainder when 3^n is divided [#permalink]

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21 Aug 2017, 08:16
Bunuel wrote:
If n is a positive integer, what is the remainder when 3^n is divided by 10?

(1) n is a multiple of 8
(2) n is a multiple of 12

The question asks what is the value, so we need to know whether a definite value is possible here.

The remainder when any number is divided by 10 will be its Units Digit. So we need to find the unit's digit of integer $$3^n$$

Statement 1: Let $$n = 8k$$. so $$3^n = 3^{8k} = (3^4)^{2k}$$
$$3^4 = 81$$, so units digit will $$1$$ and $$1$$ raised to any power will be $$1$$. So the unit's digit of $$3^n$$ will $$1$$, hence the remainder when divided by $$10$$ will be $$1$$. Sufficient

Statement : Let $$n = 12q$$. so $$3^n = 3^{12q} = (3^4)^{3q}$$
$$3^4 = 81$$, so units digit will $$1$$ and $$1$$ raised to any power will be $$1$$. So the unit's digit of $$3^n$$ will $$1$$, hence the remainder when divided by $$10$$ will be $$1$$. Sufficient

Option $$D$$
If n is a positive integer, what is the remainder when 3^n is divided   [#permalink] 21 Aug 2017, 08:16
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