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MATH 135 Fall 2021: Written Assignment 8

Q1. (5 marks) If possible, solve the following linear congruences.

(a) (2 marks) 1320x ≡ −34 (mod 2431)

(b) (3 marks) 1320x ≡ −33 (mod 2431)

Q2. (5 marks) Let m ∈ N. Let a, b and k be integers where m ∤ k. Prove or disprove each of the following statements.

(a) (2 marks) {x ∈ Z : ax ≡ b (mod m)} ⊆ {x ∈ Z : akx ≡ bk (mod m)}

(b) (3 marks) {x ∈ Z : akx ≡ bk (mod m)} ⊆ {x ∈ Z : ax ≡ b (mod m)}

Q3. (4 marks) Prove that a positive integer is divisible by 8 if and only if the number formed by its last three digits is divisible by 8.

Example: The number 12344 is divisible by 8, because 344 is divisible by 8. Also, the number 12345 is not divisible by 8, because 345 is not divisible by 8.

Q4. (5 marks) Let p and q be primes.

(a) (2 marks) Prove that p = q or gcd(p, p + q) = 1.

(b) (3 marks) Prove that p + q = (p − q)³  if and only if p = 5 and q = 3.

     Hint: Work modulo p + q.

Q5. (5 marks)

(1) Find the remainder when f(x) = x¹⁰⁰ − 1 is divided by (x + 1).

(2) Determine with explanation which of the following polynomials is divisible by (x + 1).

a. x² + x + 1

b. x² − 2x + 1

c. x⁹⁸ − x⁹⁷ + · · · − x

d. x⁹⁹ − x⁹⁸ + x⁹⁷ − x⁹⁶ + · · · + x