Exercise 1 (8 points). Prove that, for all n = N, (Hint:) you may try by...

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Exercise 1 (8 points). Prove that, for all n = N, (Hint:) you may try by induction. 5|(n-n). Exercise 2 (9 points). Solve the following systems of congruences. (a) (b) x = 6 (mod 10) x = 11 (mod 15) x = 3 (mod 21) x = 11 (mod 14) (c) x = 10 (mod 12) 2 = 16 (mod 18) x = 10 (mod 15) Exercise 3 (9 points). Compute the quotient and remainder of the polynomial division between the following pairs of polynomials in Q[x]. (a) 2x1, x22x+1. (b) 6x43x23, 2x3 + 1. (c) 3x4, x-1. 1 Exercise 1 (8 points). Prove that, for all n = N, (Hint:) you may try by induction. 5|(n-n). Exercise 2 (9 points). Solve the following systems of congruences. (a) (b) x = 6 (mod 10) x = 11 (mod 15) x = 3 (mod 21) x = 11 (mod 14) (c) x = 10 (mod 12) 2 = 16 (mod 18) x = 10 (mod 15) Exercise 3 (9 points). Compute the quotient and remainder of the polynomial division between the following pairs of polynomials in Q[x]. (a) 2x1, x22x+1. (b) 6x43x23, 2x3 + 1. (c) 3x4, x-1. 1