What is (x^2022) modulo (x^2 + 1) ?

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henh...@gmail.com

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Sep 27, 2022, 5:42:31 PMSep 27
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When you divide (x^2022) by (x^2 + 1) , what is the remainder ?


until recently, i'd not encountered problems of this type.


( x^100 ) modulo (x + 1)

( x^100 ) modulo (x^2 + 1)

( x^2022 ) modulo (x^3 + 1)

Mike Terry

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Sep 27, 2022, 6:11:24 PMSep 27
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(t - 1) ≡ -1 [mod t]

so (t - 1)^m ≡ (-1)^m ≡ +1 [mod t] (if m even)
≡ -1 [mod t] (if m odd)

This answers all your questions:

> When you divide (x^2022) by (x^2 + 1) , what is the remainder ?

x^2022 = (x^2)^1011, so taking t = x^2 + 1 , m = 1011, the answer is -1
(or x^2, assuming we want a positive remainder)

>
>
> until recently, i'd not encountered problems of this type.
>
>

Similarl approach gives...

> ( x^100 ) modulo (x + 1)>

+1

> ( x^100 ) modulo (x^2 + 1)>

+1

> ( x^2022 ) modulo (x^3 + 1)

+1


Mike.

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