BMath Bachelor of Mathematics Number Theory and Discrete Mathematics 1

Question 1

Which of the following best describes a prime number?

Accepted Answer

A number greater than 1 with no positive divisors other than 1 and itself

Answer

A prime number is a natural number greater than 1 whose only positive divisors are 1 and itself.

Question 2

The greatest common divisor gcd(48, 36) equals:

Accepted Answer

12

Answer

Using the Euclidean algorithm: gcd(48, 36) = gcd(36, 12) = gcd(12, 0) = 12.

Question 3

Fermat's Little Theorem states that if p is prime and gcd(a, p) = 1, then:

Accepted Answer

a^(p−1) ≡ 1 (mod p)

Answer

Fermat's Little Theorem: a^(p−1) ≡ 1 (mod p) for prime p not dividing a, a foundation of modular exponentiation.

Question 4

How many integers from 1 to 10 are coprime to 10?

Accepted Answer

4

Answer

φ(10) = 10·(1 − 1/2)·(1 − 1/5) = 4; the integers 1, 3, 7, 9 are coprime to 10.

Question 5

The Fundamental Theorem of Arithmetic states that every integer n > 1:

Accepted Answer

Has a unique prime factorization

Answer

Every integer greater than 1 can be uniquely expressed as a product of prime numbers (up to order of factors).

BMath Bachelor of Mathematics Practice Test