show that for all integers x and y (x-y)3=x3-y3(mod3)

5x² + 3x = 3x = 3(mod17) X = (mod17)

Find the general solution of the recursive relation an = 2an-1-12an-2 for every integer n > 1. None of these. ana(12) + B(-12)" 11 = a ( 1² ) " + B (²²) " 2 O an = a (1 + √13)" + ß(1 − √13)" O an = a (1 + √14)" + (1 - √14)" an

2x² + 3x = 3x = 14(mod19) X = X (mod19)

Show that 2x = 1 (Mod 10) has NO solutions. Show work

Give an example of a pair of numbers that can be used to generate modulo residues to represent the numbers in Z21

6. Give the addition and multiplication tables for nonconstant polynomials in Z3[x]/(x² + x+2). (i.e. just the linear polynomials x, x+1, x+2, 2x, 2x + 1, 2x + 2) Hint: By the mod x² = -x-2 (mod x² + x + 2) = 2x + 1 (mod x² + x + 2).

11. Determine a general rule to help decide whether x I x - a" and x" + a". a and x + a are factors of