BMath Bachelor of Mathematics Number Theory and Discrete Mathematics 1 — Questions and Answers
Question 1: Which of the following best describes a prime number?
- A number divisible by 2
- A number greater than 1 with no positive divisors other than 1 and itself (Correct answer)
- An even number greater than 2
- A number that is the square of an integer
Correct answer: A number greater than 1 with no positive divisors other than 1 and itself
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:
- 6
- 12 (Correct answer)
- 18
- 9
Correct answer: 12
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:
- a^p ≡ a (mod p)
- a^(p−1) ≡ 1 (mod p) (Correct answer)
- a² ≡ 1 (mod p)
- p ≡ 1 (mod a)
Correct answer: a^(p−1) ≡ 1 (mod p)
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?
- 2
- 4 (Correct answer)
- 5
- 6
Correct answer: 4
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:
- Can be written as a sum of two primes
- Has a unique prime factorization (Correct answer)
- Is either prime or a perfect square
- Has at most two prime factors
Correct answer: Has a unique prime factorization
Every integer greater than 1 can be uniquely expressed as a product of prime numbers (up to order of factors).
Question 6: In modular arithmetic, 17 mod 5 equals:
- 1
- 2 (Correct answer)
- 3
- 4
Correct answer: 2
17 = 3×5 + 2, so 17 mod 5 = 2, the remainder after dividing 17 by 5.
BMath Bachelor of Mathematics Number Theory and Discrete Mathematics 1
Which of the following best describes a prime number?