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

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

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New post 21 Aug 2017, 05:15
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If n is a positive integer, what is the remainder when 3^n is divided [#permalink]

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New post 21 Aug 2017, 06:42
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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New post 21 Aug 2017, 07:16
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, 07:16
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