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What is the remainder of 3^19 when divided by 10? [#permalink]
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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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I got D.
You can use the Mod function to solve this kind of problem.



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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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D it is...
3 ^ 16 will give remainder 1
3 ^ 3 will give remainder 7
so 7 * 1= 7



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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]
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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 103?



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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 103? 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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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



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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 Answer = D
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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?
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