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505-555 (Easy)|   Exponents|   Remainders|      
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If n is a positive integer, what is the remainder when 3^n is divided 21 Aug 2017, 06:15
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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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D
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If n is a positive integer, what is the remainder when 3^n is divided 21 Aug 2017, 07:42
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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 21 Aug 2017, 08:16
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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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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 \(D\)
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If n is a positive integer, what is the remainder when 3^n is divided 06 Aug 2020, 13:40
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We need to find the units digit of 3^n. This will enable us to find what the remainder is going to be when 3^n is divided by 10.

Find the units digits of 3. This cycle is going to repeat.
3 = [3 (from 3^1), 9 (from 3^2), 7 (3^3),1 (3^4)]

(1) The units digit for a multiple of 8 is going to be 1 (go through cycle 2 times). Remainder will be 1. Suff.

(2) Similarly, the units digit for a multiple of 12 is going to be 1 (go through cycle 3 times). Remainder will be 1. Suff.

Answer: D
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If n is a positive integer, what is the remainder when 3^n is divided 14 Oct 2020, 20:55
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If n is a positive integer, what is the remainder when 3^n is divided by 10?

The way I see it...

\(3^n/10\) can have the following remainders, repeated.

[7, 1, 3, 9]

This is the case because we understand the units digit pattern that occurs when 3 is put to an integer power from 1 -> inf is:

[3, 9, 7, 1]

We can then subtract 10 from each item in the list to retrieve what the candidate remainders would be.

10 - [3, 9, 7, 1] = [7, 1, 3, 9].

If we can pinpoint the pattern, we have our answer.

(1) n is a multiple of 8

n, being a multiple of 8, means that the remainder will be 9.
\(8/4 = 2\), meaning that the pattern will repeat itself fully twice. Sufficient

(2) n is a multiple of 12

n, being a multiple of 12, means that the remainder will also be 9.
\(12/4 = 3\), meaning that the pattern will repeat itself fully three times. Sufficient
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If n is a positive integer, what is the remainder when 3^n is divided 06 Oct 2024, 06:40
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What is the remainder when 3^n is divided by 10?

Theory: Remainder of a number by 10 is same as the unit's digit of the number

(Watch this Video to Learn How to find Remainders of Numbers by 10)

Using Above theory Remainder of \(3^{n} \) by 10 = Unit's digit of \( 3^{n} \)

Now to find the unit's digit of \( 3^{n} \) , we need to find the pattern / cycle of unit's digit of power of 3 and then generalizing it.

Unit's digit of \(3^1\) = 3
Unit's digit of \(3^2\) = 9
Unit's digit of \(3^3\) = 7
Unit's digit of \(3^4\) = 1
Unit's digit of \(3^5\) = 3

So, units' digit of power of 3 repeats after every \(4^{th}\) number.
=> We need to divided n by 4 and check what is the remainder

STAT 1: n is a multiple of 8

=> n is a multiple of 8, so it will be a multiple of 4 too
=> n when divided by 4 will give 0 remainder
=> Units' digit of 3^n will be same as units' digit of \(3^{Cycle}\) = Units' digit of 3^4 = 1
=> Remainder of 3^n by 10 = 1
=> SUFFICIENT

STAT 2: n is a multiple of 12

=> n is a multiple of 12, so it will be a multiple of 4 too
=> n when divided by 4 will give 0 remainder
=> Units' digit of 3^n will be same as units' digit of \(3^{Cycle}\) = Units' digit of 3^4 = 1
=> Remainder of 3^n by 10 = 1
=> SUFFICIENT

So, Answer will be D
Hope it helps!

MASTER How to Find Remainders with 2, 3, 5, 9, 10 and Binomial Theorem

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