Determine whether each of the statements is true or false

Question

Question: Complete solutions with full working final solutions to be typed up on words or pdf.

Q1

(a) Using unique prime factorisation, find gcd(12100, 4488).

(b) Using the Euclidean algorithm, find gcd(460350, 3315).

(c) Let a, b, d ∈ N. Prove that, if d is a common divisor of a and b , then ab/d is a common multiple of a and b .

(d) Let a, b, m ∈ N. Prove that, if a | m and b | m, then lcm(a, b) | m.

Hint: There are at least two ways to do this. One solution uses unique prime factorisation, and one solution uses the quotient-remainder theorem.

Q2 Determine whether each of the following statements is true or false. If true, give a proof. If false, give a counterexample.

(a) Let x ∈ R and let n ∈ N. Then n.[x] =[n.x]

(b) The product of two irrational numbers is always irrational.

(c) The sum of a rational number and an irrational number is always irrational.

(d) Let d, n ∈ N. If d | n and d > 1, then d † (2n + 1).

(e) Let d, n ∈ N, and let n be even. If d | n and d > 1, then d † ( n + 1).

Q3 A perfect cube is an integer of the form n³ where n ∈ Z.

(a) What possible remainders do perfect cubes leave when divided by 7?

(b) What possible remainders do perfect cubes leave when divided by 13?

(c) Show that there are no integer solutions a, b ∈ Z to the equation a³ = b⁴ + 6.

Hint: Your answers to (a) and/or (b) might come in useful.

Q4 (a) True or false? For all n ∈ Z, if 10 | n², then 10 | n. If true, give a proof. If false, give a counterexample.

(b) True or false? For all n ∈ Z, if 12 | n², then 12 | n. If true, give a proof. If false, give a counterexample.

(c) Prove that √15 is irrational.

(d) Prove that ³√4 is irrational.