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08 Dec 2021, 07:15
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E
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Difficulty:
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43% (01:50) correct
57%
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wrong
based on 289
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What is the remainder when \(81^{82}\) is divided by 17 ?
(A) 1
(B) 4
(C) 13
(D) 15
(E) 16
Are You Up For the Challenge: 700 Level Questions
(A) 1
(B) 4
(C) 13
(D) 15
(E) 16
Are You Up For the Challenge: 700 Level Questions
08 Dec 2021, 09:04
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We are asked to find the remainder of \(\frac{(81) ^ {82} }{ 17 } \)
81 can be expressed as (17 * 4 ) + 13.
So essentially, we need to find the remainder of \(\frac{(13) ^ {82} }{ 17 } \), as 17 * 4 will leave a remainder of 0.
\(\frac{(13 * 13) ^ {81} }{ 17 } \) leaves a remainder of \(\frac{(-1) ^ {81} }{ 17 } \), which is equal to -1. However as remainders cannot be negative, the net remainder is 17 - 1 = 16.
IMO - E
81 can be expressed as (17 * 4 ) + 13.
So essentially, we need to find the remainder of \(\frac{(13) ^ {82} }{ 17 } \), as 17 * 4 will leave a remainder of 0.
\(\frac{(13 * 13) ^ {81} }{ 17 } \) leaves a remainder of \(\frac{(-1) ^ {81} }{ 17 } \), which is equal to -1. However as remainders cannot be negative, the net remainder is 17 - 1 = 16.
IMO - E
15 Sep 2022, 10:13
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We need to find what is the remainder when \(81^{82}\) is divided by 17
\(81^{82}\) = \((85 - 4)^{82}\) = \((17*5 - 4)^{82}\)
using binomial theorem if we open this then all terms except the last term will be a multiple of 17 => Remainder of all the terms except the last term with 17 will be zero
=> Remainder will be same as the remainder of the last term = \((-4)^{82}\) = \((-4)^{2*41}\) = \(((-4)^2)^{41}\) = \(16^{41}\) = \((17 -1)^{41}\)
using binomial theorem if we open this then all terms except the last term will be a multiple of 17 => Remainder of all the terms except the last term with 17 will be zero
=> Remainder will be same as the remainder of the last term = \((-1)^{41}\) = Remainder of -1 by 17 = 16
So, Answer will be E
Hope it helps!
Watch the following video to learn the Basics of Remainders
\(81^{82}\) = \((85 - 4)^{82}\) = \((17*5 - 4)^{82}\)
using binomial theorem if we open this then all terms except the last term will be a multiple of 17 => Remainder of all the terms except the last term with 17 will be zero
=> Remainder will be same as the remainder of the last term = \((-4)^{82}\) = \((-4)^{2*41}\) = \(((-4)^2)^{41}\) = \(16^{41}\) = \((17 -1)^{41}\)
using binomial theorem if we open this then all terms except the last term will be a multiple of 17 => Remainder of all the terms except the last term with 17 will be zero
=> Remainder will be same as the remainder of the last term = \((-1)^{41}\) = Remainder of -1 by 17 = 16
So, Answer will be E
Hope it helps!
Watch the following video to learn the Basics of Remainders
