# Can the coefficients of these two pairs of polynomials be reduced? If yes, how do you do it with sagemath?

Can the coefficients of these two pairs of polynomials be reduced?

If yes, how do you do it with sagemath?

(8223*b+6341)*(8221*a+6339) mod 67586227 = 0

(8213-8*b)*(8215-8*a) = 67586227


or

(8223*b+6341)*(8215-8*a) mod 67586227 = 0

(8221*a+6339)*(8213-8*b) mod 67586227 = 0

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One option is to work with polynomials, say with integer coefficients:

sage: S.<a,b> = ZZ[]
sage: f = (8223*b+6341)*(8221*a+6339)


Now reduce f mod 67586227:

sage: R = IntegerModRing(67586227)
sage: f
67601283*a*b + 52129361*a + 52125597*b + 40195599
sage: f.change_ring(R)
15056*a*b + 52129361*a + 52125597*b + 40195599

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@John Palmieri there were errors, now it is corrected excuse me

( 2022-09-27 11:02:06 +0200 )edit

I showed how to reduce the first one; you can do the same with all of the others.

( 2022-09-27 16:41:36 +0200 )edit