Lecture 2

Question 1

Define divisibility

Answer 1

We say d divides n (d∣n) whenever n = cd for some c. If d does not divide n, then we write d ∤ n

Question 2

Give the properties of divisibility

Answer 2

1. d∣m and d∣n ⇒ d∣(am + bn).
2. ad∣an and a ≠ 0 ⇒ d∣n.
3. d∣n and n ≠ 0 ⇒ ∣d∣ ≤ ∣n∣.
4. d∣n and n∣d ⇒ ∣d∣ = ∣n∣.
5. d∣n and nd ≠ 0 ⇒ (n/d) ∣n.

Question 3

Define a common divisor

Answer 3

If d∣a and d∣b, then d is called a common divisor of a and b.

Question 4

What are the three properties of the greatest common divisor

Answer 4

1. d ≥ 0.
2. d∣a and d∣b.
3. e∣a and e∣b ⇒ e∣d.

Question 5

Define the greatest common divisor

Answer 5

The number d which satisfies the properties seen in the previous Theorem is called the greatest common divisor of a and b, which is denoted by gcd(a, b) or (a, b) or aDb

Question 6

Euclid’s lemma

Answer 6

If a∣bc and gcd(a, b) = 1, then a∣c.

Question 7

Define Prime numbers

Answer 7

An integer n is called prime if n > 1 and if the only positive divisors of n are 1 and n itself. If n > 1 and is not a prime, then n is composite

Question 8

Give theorem 9, the almagamation of several theorems about prime numbers

Answer 8

1. Every integer n > 1 is a prime number or a product of prime numbers.
2. There are infinitely many prime numbers.
3. If p is a prime and p ∤ a, then (p, a) = 1.
4. If p is a prime and p∣ab, then p∣a or p∣b. More generally if p∣a1 . . . an, then p divides at least one of the factors.
5. (Fundamental Theorem of Arithmetic) Every integer n > 1 can be represented as a product of prime factors in only one way, apart from the order of the factors.

Question 9

Does the following converge
Sum_(n=1)^infinity 1/p_n

Answer 9

Diverges

Question 10

Given a and b with b > 0, does there existsa unique pair of integers q and r such that a = bq + r with 0 ≤ r < b.

Answer 10

Yes. Moreover r = 0 if and only if b∣a

Question 11

What is the Euclidean algorithm

Answer 11

Given positive integers a and b where b ∤ a. Let r0 = a and r1 = b, and apply the division repeatedly to obtain a set of remainders r2, . . . , rn+1 defined succesively by the relations:

See equation sheet

Then r_n = gcd(a,b)