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What is the remainder of 3^19 when divided by 10?

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XxxyyY
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What is the remainder of 3^19 when divided by 10? [#permalink] New post   27 May 2006, 22:53
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What is the remainder of 3^19 when divided by 10?

A. 0
B. 1
C. 5
D. 7
E. 9
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D
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Re: How do you solve? [#permalink] New post   07 Feb 2014, 17:15
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Luning1 wrote:
what is the remainder when 3^19 is divided by 10

The answer is 7 but I don't understand why. Is it as simple as 10-3?


Just look at how power of 3 works..

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 =243
.....

You'd notice that unit digit repeats itself after every 4 numbers.. In other words, 3^19 will have same unit number as 3^15, or 3^11 or 3^7 or 3^3....Which tells us, 3^19 has unit digit as 7..

Now, we know that any number with unit digit as 7 when divided by 10 will give u 7....

PS: Wrong section... Moving it to PS..
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[#permalink] New post   27 May 2006, 23:17
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3^19=3^4*3^4*3^4*3^4*3^3
3^4=81/10 has rmainder 1
3^3=27/10 has remainder 7
1*1*1*1*7=7
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[#permalink] New post   27 May 2006, 23:17
I got D.
You can use the Mod function to solve this kind of problem.
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[#permalink] New post   27 May 2006, 23:21
D it is...

3 ^ 16 will give remainder 1
3 ^ 3 will give remainder 7

so 7 * 1= 7
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[#permalink] New post   31 May 2006, 18:12
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I am getting 7 - D

brute force to find the pattern in this case 39713971 and then just count to the 19th power we get 7
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How do you solve? [#permalink] New post   07 Feb 2014, 17:06
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what is the remainder when 3^19 is divided by 10

The answer is 7 but I don't understand why. Is it as simple as 10-3?
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Re: How do you solve? [#permalink] New post   07 Feb 2014, 19:29
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what is the remainder when 3^19 is divided by 10?

Units digit of 3^n (n = 1,2,3....) follows the following pattern: 3,9,7,1,3,9,7,1...
When divided by 10, the remainders would be 3,9,7,1,3,9,7,1...
3^19/10 -> remainder = 7
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Re: What is the remainder of 3^19 when divided by 10? [#permalink] New post   10 Sep 2014, 21:52
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we are interested in last digit of 3^19, so we should find circulation

3*1=3
3*2=9
3*3=7
3*4=1
3*5=3
circulation is 4, that means 19/4=4 with 3 remainder, so 7 is last digit. Dividing 10 gives us 7/10, so 7 is remainder

D
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What is the remainder of 3^19 when divided by 10? [#permalink] New post   12 Sep 2014, 11:48
I got D, and this is how I solved:
I looked for patterns:
^2 - units digit 9
^3 - units digit 7
^4 - units digit 1
^5 - units digit 3

hence, we can see that when raised to a power which is multiple of 4, the units digit is 1, and when to an even power not multiple of 4, the units digit is 9
and we can then see:
^16 - units digit 1, or
^18 - units digit 9
and ^19 - units digit 7

therefore, when divided by 10, the remainder must be 7
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Re: What is the remainder of 3^19 when divided by 10? [#permalink] New post   30 Sep 2014, 23:23
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For \(3^1\), remainder = 3

\(3^2\), remainder = 9

\(3^3\), remainder = 7

\(3^4\), remainder = 1

& so on........

For \(3^{19}\), remainder = 7

Answer = D
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Re: What is the remainder of 3^19 when divided by 10? [#permalink] New post   14 Nov 2015, 12:21
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X & Y wrote:
What is the remainder of 3^19 when divided by 10?

A. 0
B. 1
C. 5
D. 7
E. 9


We know \(\frac{3^4}{10}\) = Reminder 1

3^19 = \(\frac{(3^4)^4 *(3^3)}{10}\)

\(\frac{(3^4)^4}{10}\) will have remainder 1

We need to find the reminder of \(\frac{(3^3)}{10}\)

\(\frac{(3^3)}{10} = [m]\frac{(20 + 7)}{10}\) => Reminder is 7
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What is the remainder of 3^19 when divided by 10? [#permalink] New post   30 Jun 2017, 20:49
The question is essentially asking - What is the unit digit of \(3^{19}\)

Quick check on cycilicity of unit digit says

\(3^1 = 3\)

\(3^2 = 9\)

\(3^3 = 7\)

\(3^4 = 1\)

Therefore, the unit digit repeats after every 4th power of 3. For unit digit of \(3^{19}\) First find \(Remainder \ of (\frac{19}{4})= 3\). The unit digit is 3rd term in the above sequence = 7

Answer is D
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Re: What is the remainder of 3^19 when divided by 10? [#permalink] New post   01 Sep 2020, 02:33
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\(3^{19}\) = \((3^2)^{9}\) * 3

=> \(\frac{(3^2) }{ 10}\) -> remainder = -1

=> \((-1)^9\) = -1

=>\( \frac{3 }{ 10}\) -> remainder = -7

- 1 * - 7 = 7

Answer D
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Re: What is the remainder of 3^19 when divided by 10? [#permalink] New post   07 Sep 2022, 12:01
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What is the remainder of \(3^{19}\) when 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 \(3^{19}\) by 10 = unit's digit of \(3^{19}\)

Now to find the unit's digit of \(3^{19}\), 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, unit's digit of power of 3 repeats after every \(4^{th}\) number.
=> We need to divided 19 by 4 and check what is the remainder
=> 19 divided by 4 gives 3 remainder

=> \(3^{19}\) will have the same unit's digit as \(3^3\) = 7

So, Answer will be D
Hope it helps!

Watch the following video to learn the Basics of Remainders

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Re: What is the remainder of 3^19 when divided by 10? [#permalink] New post   02 Feb 2023, 14:05
I started counting from 3^0 and I got 9 as an answer, how do I know if I have to start from 3^0 or 3^1?

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Re: What is the remainder of 3^19 when divided by 10? [#permalink]
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