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The remainder when the difference of 3191^2020 and 3159^2020 is

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Afroditee
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The remainder when the difference of 3191^2020 and 3159^2020 is [#permalink] New post   Updated on: 12 Nov 2022, 10:07
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What is the remainder when the difference of \(3191^{2020}\) and \(3159^{2020}\) is divided by 16 ?

A. 0

B. 3

C. 4

D. 6

E. 7
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A

Originally posted by Afroditee on 12 Nov 2022, 10:05.
Last edited by Bunuel on 12 Nov 2022, 10:07, edited 1 time in total.
Formatted.
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IanStewart
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Re: The remainder when the difference of 3191^2020 and 3159^2020 is [#permalink] New post   12 Nov 2022, 10:38
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The numbers don't really matter here; all that matters is that the bases are odd, and the exponents are multiples of 4. So we really just have odd^4 - odd^4 = (odd^2 + odd^2)(odd^2 - odd^2) = (odd^2 + odd^2)(odd + odd)(odd - odd), and all three of those factors are even, so their product will be a multiple of 8. A multiple of 8 can only have a remainder of 0 or 8 when you divide by 16, and 0 is the only possibility among the choices.

It's not hard to prove that odd^2 - odd^2 is actually always a multiple of 8 (which is why the answer is zero here), but we don't need to bother doing that for this question with these answer choices.
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Re: The remainder when the difference of 3191^2020 and 3159^2020 is [#permalink] New post   19 Nov 2022, 13:09
IanStewart wrote:
The numbers don't really matter here; all that matters is that the bases are odd, and the exponents are multiples of 4. So we really just have odd^4 - odd^4 = (odd^2 + odd^2)(odd^2 - odd^2) = (odd^2 + odd^2)(odd + odd)(odd - odd), and all three of those factors are even, so their product will be a multiple of 8. A multiple of 8 can only have a remainder of 0 or 8 when you divide by 16, and 0 is the only possibility among the choices.

It's not hard to prove that odd^2 - odd^2 is actually always a multiple of 8 (which is why the answer is zero here), but we don't need to bother doing that for this question with these answer choices.

Could you give a more simple solution? Thanks
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Re: The remainder when the difference of 3191^2020 and 3159^2020 is [#permalink] New post   19 Nov 2022, 20:21
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Afroditee wrote:
What is the remainder when the difference of \(3191^{2020}\) and \(3159^{2020}\) is divided by 16 ?

A. 0

B. 3

C. 4

D. 6

E. 7



IanStewart follows a very apt logical approach.

But if you are looking for a quick answer, then remember these rules/facts/deductions
\(a^n-b^n\) is always divisible by a-b
So, \(3191^{2020}\) - \(3159^{2020}\) is divisible by 3191-3159 or 32, and so by 16 also.
Thus, remainder is 0.

Other points that one could remember are
\(a^n-b^n\) is divisible by a+b if n is even
\(a^n+b^n\) is divisible by a+b if n is even


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Re: The remainder when the difference of 3191^2020 and 3159^2020 is [#permalink] New post   20 Nov 2022, 02:45
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What is the remainder when the difference of 3191^2020 and 3159^2020 is divided by 16 ?

A. 0

B. 3

C. 4

D. 6

E. 7

Solution:-
we can solve this with the help of highest power method
so we can use 16= 2^4= implies highest power is 5
and 2020 is divided by 5 implies reminder is 0
so in case of both 3191^2020 and 3159^2020 difference
we can say that each give reminder 0 in so reminder is Zero
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Re: The remainder when the difference of 3191^2020 and 3159^2020 is [#permalink]
20 Nov 2022, 02:45
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