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

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Manager
Joined: 25 May 2006
Posts: 227
What is the remainder of 3^19 when divided by 10? [#permalink]

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27 May 2006, 22:53
3
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Question Stats:

75% (01:36) correct 25% (01:03) wrong based on 239 sessions

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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
[Reveal] Spoiler: OA
VP
Joined: 07 Nov 2005
Posts: 1118
Location: India

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27 May 2006, 23:17
I got D.
You can use the Mod function to solve this kind of problem.
Director
Joined: 13 Nov 2003
Posts: 789
Location: BULGARIA

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27 May 2006, 23:17
1
KUDOS
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
Director
Joined: 10 Oct 2005
Posts: 526
Location: US

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27 May 2006, 23:21
D it is...

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

so 7 * 1= 7
Manager
Joined: 11 Oct 2005
Posts: 94

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31 May 2006, 18:12
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
Intern
Joined: 07 Feb 2014
Posts: 3

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07 Feb 2014, 17:06
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?
Director
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Schools: Olin - Wash U - Class of 2015
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Re: How do you solve? [#permalink]

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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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Intern
Joined: 01 Aug 2006
Posts: 35
Re: How do you solve? [#permalink]

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07 Feb 2014, 19:29
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
Director
Joined: 23 Jan 2013
Posts: 585
Schools: Cambridge'16
Re: What is the remainder of 3^19 when divided by 10? [#permalink]

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10 Sep 2014, 21:52
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]

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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]

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30 Sep 2014, 23:23
For $$3^1$$, remainder = 3

$$3^2$$, remainder = 9

$$3^3$$, remainder = 7

$$3^4$$, remainder = 1

& so on........

For $$3^{19}$$, remainder = 7

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

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14 Nov 2015, 03:06
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Re: What is the remainder of 3^19 when divided by 10? [#permalink]

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14 Nov 2015, 12:21
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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Re: What is the remainder of 3^19 when divided by 10?   [#permalink] 14 Nov 2015, 12:21
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