BMath Bachelor of Mathematics Number Theory and Discrete Mathematics 3

Question 1

Wilson's Theorem states that p is prime if and only if:

Accepted Answer

(p − 1)! ≡ −1 (mod p)

Answer

(p − 2)! ≡ 1 (mod p)

Answer

(p + 1)! ≡ 1 (mod p)

Answer

(p − 1)! ≡ 0 (mod p)

Question 2

A bijection between two sets A and B implies:

Accepted Answer

|A| = |B|

Answer

|A| < |B|

Answer

|A| > |B|

Answer

A ⊆ B

Question 3

The Pigeonhole Principle states that if n + 1 objects are placed into n containers, then:

Accepted Answer

At least one container has at least 2 objects

Answer

At least one container is empty

Answer

All containers have the same number of objects

Answer

Exactly one container has 2 objects

Question 4

A relation R on a set S is an equivalence relation if it is:

Accepted Answer

Reflexive, symmetric, and transitive

Answer

Reflexive and transitive only

Answer

Symmetric and transitive only

Answer

Antisymmetric and transitive

Question 5

The Chinese Remainder Theorem guarantees a unique solution modulo m₁m₂ when gcd(m₁, m₂) equals:

Accepted Answer

1

Answer

m₁

Answer

m₂

Answer

m₁ + m₂

Question 6

In a simple graph, the sum of all vertex degrees equals:

Accepted Answer

Twice the number of edges

Answer

The number of vertices

Answer

The number of edges

Answer

Half the number of vertices