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Quizzes > Mathematics & Statistics

Number Theory Quiz

Free Practice Quiz & Exam Preparation

Difficulty: Moderate
Questions: 15
Study OutcomesAdditional Reading
3D voxel art illustrating concepts from the Number Theory course

This Number Theory quiz helps you practice core skills and find gaps before an exam. Work through 15 quick questions on divisibility, primes, congruences, factorization, quadratic residues, arithmetic functions, and primitive roots. Get clear results with brief notes and pointers for further study, so you know what to review next.

What does it mean for an integer a to divide an integer b?
b is always greater than a.
There exists an integer k such that b = a * k.
b divides a.
a is a factor of b, but not vice versa.
An integer a divides b if there exists some integer k such that b equals a multiplied by k. This definition is fundamental in determining divisibility in number theory.
Which statement correctly defines a prime number?
A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself.
A prime number is any even number greater than 2.
A prime number is an integer with exactly three positive divisors.
A prime number is any integer that is odd.
A prime number is defined as an integer greater than 1 that has exactly two distinct positive divisors. This clear-cut definition distinguishes prime numbers from composite numbers.
What is the greatest common divisor (gcd) of 14 and 21?
3
21
7
14
The greatest common divisor (gcd) is the largest number that divides both given numbers evenly. Since 7 divides both 14 and 21, the gcd is 7.
Which of the following is a divisor of 17?
11
2
17
5
Since 17 is a prime number, its only divisors are 1 and itself. Among the options provided, 17 is the divisor that satisfies this property.
Which of the following congruence statements is true?
15 ≡ 3 (mod 4) because both leave a remainder of 3 when divided by 4.
15 ≡ 3 (mod 4) because 15 and 3 are both multiples of 4.
15 ≡ 3 (mod 4) because the sum of 15 and 3 is divisible by 4.
15 ≡ 3 (mod 4) because 15 - 3 is 10 and 10 is a multiple of 4.
Two numbers are congruent modulo 4 if they yield the same remainder when divided by 4. Since both 15 and 3 give a remainder of 3 upon division by 4, the congruence statement is true.
What does Fermat's Little Theorem state?
If a and p are relatively prime, then a^p ≡ 1 (mod p).
If p is a prime and a is not divisible by p, then a^(p-1) ≡ 1 (mod p).
If p is prime, then a^p ≡ a (mod p) for every integer a.
If a is an integer, then a^(p-1) ≡ a (mod p) for any prime p.
Fermat's Little Theorem asserts that if p is a prime number and a is not a multiple of p, then raising a to the power of (p-1) will yield a number that is congruent to 1 modulo p. This theorem is fundamental in simplifying computations in modular arithmetic and in cryptographic applications.
What does the Legendre symbol (a/p) represent?
It is defined only when a and p are relatively prime.
It equals 1 if a is a quadratic residue modulo p, -1 if a is a non-residue, and 0 if p divides a.
It only takes the values 0 and 1 depending on whether a is divisible by p.
It represents the number of solutions to the equation x² ≡ a (mod p).
The Legendre symbol is a notational tool used to succinctly indicate whether an integer a is a quadratic residue modulo a prime p. Its values capture whether a has a square root modulo p, which is crucial in studying quadratic congruences.
According to the Law of Quadratic Reciprocity, if p and q are distinct odd primes, then:
The sum of (p/q) and (q/p) equals (-1)^((p-1)(q-1)/4).
The Legendre symbols (p/q) and (q/p) are always equal.
Both (p/q) and (q/p) are always 1 for any odd primes p and q.
The product of (p/q) and (q/p) equals (-1)^((p-1)(q-1)/4).
The Law of Quadratic Reciprocity provides a deep connection between the quadratic residues of two different odd primes. It indicates that the product of the Legendre symbols (p/q) and (q/p) is given by (-1) raised to the power ((p-1)(q-1)/4), revealing a beautiful symmetry in number theory.
Which of the following best describes a primitive root modulo n?
A solution to the congruence g^k ≡ 0 (mod n) for some k.
An integer for which g² is congruent to 1 modulo n.
Any integer less than n that is relatively prime to n.
An integer g such that the powers of g generate all the units modulo n.
A primitive root modulo n is an integer whose successive powers produce every number that is coprime to n under modular arithmetic. Understanding primitive roots is key to studying the multiplicative structure of the group of units modulo n.
Which arithmetic function counts the number of positive integers up to n that are coprime to n?
The sum-of-divisors function, σ(n).
The Möbius function, μ(n).
Euler's totient function, φ(n).
The divisor function, d(n).
Euler's totient function, denoted as φ(n), measures the count of integers up to n that are relatively prime to n. It plays a central role in various theorems in number theory including Euler's theorem and its applications in cryptography.
What is a necessary condition for the linear Diophantine equation ax + by = c to have an integer solution?
c must be divisible by the greatest common divisor of a and b.
a, b, and c must all be prime numbers.
c must be greater than both a and b.
a and b must be relatively prime.
For the linear Diophantine equation ax + by = c to admit integer solutions, it is essential that c is divisible by gcd(a, b). This condition arises from the fundamental properties of integer linear combinations.
Which statement correctly describes a continued fraction representation?
It always gives a terminating decimal expansion.
It expresses a number as an integer plus a fraction whose denominator is another number, often resulting in periodic sequences for quadratic irrationals.
It is used to express complex numbers in a simplified form.
It represents a number solely as a finite sum of fractions.
A continued fraction expresses a number by isolating its integer part and then representing the remaining fractional part as the reciprocal of another number. This method is especially useful in approximating irrational numbers and revealing periodic behavior in quadratic irrationals.
What is a Farey sequence of order n?
A sequence of fractions that best approximate irrational numbers.
The set of all fractions with both numerator and denominator between 0 and n.
An ascending sequence of irreducible fractions between 0 and 1 with denominators less than or equal to n.
A descending sequence of fractions with denominators equal to n.
A Farey sequence of order n lists in increasing order all the fractions between 0 and 1 that are in their lowest terms with denominators not exceeding n. This sequence is a useful tool in understanding the distribution of rational numbers.
In cryptography, which number theoretic concept is primarily utilized in the RSA algorithm?
The periodicity of continued fractions.
The properties of quadratic residues.
The structure of linear Diophantine equations.
The difficulty of factoring large composite numbers into prime factors.
RSA encryption is based on the principle that while it is relatively easy to multiply two large prime numbers, factoring their product is computationally hard. This one-way function forms the backbone of RSA's security.
Which of the following best describes a linear recurrence relation with constant coefficients?
A sequence that cannot be solved using characteristic equations.
A sequence determined by the product of the previous terms.
A sequence where each term is a constant multiple of the preceding term only.
A sequence where each term is a linear combination of a fixed number of previous terms with constant multipliers.
A linear recurrence relation with constant coefficients defines a sequence in which every term is expressed as a fixed linear combination of previous terms. This property allows the use of characteristic equations to derive closed-form solutions.
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Study Outcomes

  1. Analyze divisibility rules and prime factorization techniques.
  2. Apply congruence relations to solve number theory problems.
  3. Evaluate quadratic residues and utilize quadratic reciprocity principles.
  4. Synthesize the use of arithmetic functions and primitive roots in cryptographic applications.

Number Theory Additional Reading

Embarking on a journey through number theory? Here are some top-notch resources to guide you:

  1. Dive into MIT's comprehensive course featuring lecture notes, assignments, and related resources, all tailored to deepen your understanding of number theory.
  2. Access detailed lecture notes covering topics from absolute values to global class field theory, serving as an excellent online textbook for your studies.
  3. Explore a curated list of recommended texts and resources, including classics like Niven and Zuckerman's "An Introduction to the Theory of Numbers," to supplement your learning.
  4. Delve into analytic number theory with lecture notes covering the prime number theorem, Dirichlet series, and more, complete with homework questions to test your knowledge.
  5. Download the entire course package, including lecture notes and problem sets, for offline study and a structured approach to mastering number theory.
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