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

Wilson's Theorem: p is prime iff (p − 1)! ≡ −1 (mod p), providing a primality characterization though not an efficient test.

Question 2

A bijection between two sets A and B implies:

Accepted Answer

|A| = |B|

Answer

A bijection (one-to-one and onto function) between A and B means they have the same cardinality, |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

With more objects than containers, at least one container must hold at least two objects by the Pigeonhole Principle.

Question 4

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

Accepted Answer

Reflexive, symmetric, and transitive

Answer

An equivalence relation must be reflexive (aRa), symmetric (aRb ⟹ bRa), and transitive (aRb and bRc ⟹ aRc).

Question 5

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

Accepted Answer

1

Answer

The CRT applies when the moduli are pairwise coprime (gcd = 1), ensuring the simultaneous congruences have a unique solution modulo m₁m₂.

BMath Bachelor of Mathematics Practice Test