• What is (x^2022) modulo (x^2 + 1) ?

From henhanna@gmail.com@21:1/5 to All on Tue Sep 27 14:42:30 2022
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)

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• From Mike Terry@21:1/5 to henh...@gmail.com on Tue Sep 27 23:11:17 2022
On 27/09/2022 22:42, henh...@gmail.com wrote:

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)

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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)

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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