Number Theory Questions

Number theory questions and answers are available on this page and contain various types of questions. These questions cover different types of techniques and formulas in mathematics and are particularly related to numbers. All these number theory questions will help you to understand how to solve complex problems using simple shortcuts and tricks.

What is the number theory?

Number theory is one of the elementary branches of mathematics that deals with the study of numbers (natural numbers) and properties of numbers, classification of numbers based on certain arithmetic operations.

Also read: Number theory

Number Theory Questions and Answers

1. Find the number of trailing zeros in the 100!.

Let us divide 100 by 5.

Now, 4 is less than 5. So, we stop the division here.

Also, the number of trailing zeros = 20 + 4 = 24

2. Why is 1 not considered to be prime?

From the definition of prime numbers, we can say that a number is considered to be prime if it contains two distinct factors, namely, 1 and the number itself. The number 1 is divided by 1 itself. Thus, it cannot have any other factor. Hence, 1 is not considered to be prime.

3. Find the LCM of 8, 27 and 72 using the prime factorisation method.

Prime factorisation of 8 = 2 × 2 × 2

Prime factorisation of 27 = 3 × 3 × 3

Prime factorisation of 72 = 2 × 2 × 2 × 3 × 3

LCM(8, 27, 72) = 2 × 2 × 2 × 3 × 3 × 3 = 216

Therefore, the LCM of 8, 27 and 72 is 216.

4. Write the first 10 non-zero multiples of 7.

Multiple of a number is the result obtained when the number is multiplied by integers.

The first 10 non-zero multiples of 7 are written as follows:

7 × 10 = 70

Thus, the first 10 non-zero multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63 and 70.

5. What is the value of (61 2 − 39 2 ) ÷ (51 2 − 49 2 )?

(61 2 − 39 2 ) ÷ (512 − 492)

Consider (61 2 – 39 2 )

Using the identity a 2 – b 2 = (a + b)(a – b)

(61 2 – 39 2 ) = (61 + 39) (61 – 39)

(51 2 – 49 2 ) = (51 + 49) (51 – 49)

Therefore, (61 2 − 39 2 ) ÷ (51 2 − 49 2 ) = (100 × 22)/ (100 × 2) = 11

6. What are the common factors of 48 and 54?

Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54

Here, 1, 2, 3, and 6 are common for both 48 and 54.

Thus, the common factors of 48 and 54 are 1, 2, 3 and 6.

7. Write the first three common multiples of 15 and 20.

The multiples of 15 are 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, etc.

The multiples of 20 are 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, etc.

The first three common multiples of 15 and 20 are 60, 90 and 120.

8. Prove that the sequence 2, 5, 8, 11, 14, 17,.… can never have a square number.

2, 5, 8, 11, 14, 17,.…

This is an AP with the first term a = 2 and common difference d = 5 – 2 = 3.

nth term of the given sequence is:

a n = a + (n – 1) × d

a n = 5 + (n – 1) × 3

= 5 + 3n – 3

∴ a n = 3n + 2

Let p be a natural number, such that p 2 = an

p 2 = 3n + 2

3n = p 2 – 2

n = (p 2 – 2)/3

For any integer from 0 to 9 for p, n does not appear to be an integer.

Hence, the given sequence contains no perfect squares.

9. Find all primes that can be represented as sums and differences between two primes.

Let x be a prime that can be represented both as a sum and as a difference of 2 primes.

For the given statement, we must have x > 2.

∴ x is an odd prime number.

Also, one of those prime numbers must be 2.

That means we must have x = y + 2 = z – 2, where y and z are prime numbers.

So, z = x + 2 and y = z + 2

Thus, we can say that x, y and z will be three consecutive odd primes.

As we know, there is only one such set of prime numbers, and they are 3, 5, and 7.

x = 5 = 3 + 2 = 7 – 2

Hence, there is only one prime number .i.e 5 can be represented as sums and differences between two primes.

10. What is the divisibility rule of 11? Give one example.

If the difference between the sum of digits of a number at odd places and the sum of digits of the number at even places is equal to 0 or a multiple of 11, then the number is divisible by 11.

Consider the number 719752

Sum of digits at odd places = 7 + 9 + 5 = 21

Sum of digits at even places = 1 + 7 + 2 = 10

Difference = 21 – 10 = 11 (multiple of 11)

Therefore, the number 719752 is divisible by 11.

Practice Questions on Number Theory

  • If the product of the integers w, x, y and z is 770, and if 1 < w < x < y < z, what is the value of w + z?
  • Find the GCF of 67 and 194 using the prime factorisation method.
  • Identify the number of zeros in the value of 1000!.
  • What is the divisibility rule for 7?
  • What is the largest five-digit number that is divisible by 7, 10, 15, 21 and 28?

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  • Number Theory
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Take a guided, problem-solving based approach to learning Number Theory. These compilations provide unique perspectives and applications you won't find anywhere else.

What's inside

  • Introduction
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Browse through thousands of Number Theory wikis written by our community of experts.

  • Even and Odd Numbers
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  • Rightmost Non-zero Digit of \(n!\)
  • Wilson's Theorem
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  • Is 0 even, odd, or neither?
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  • Modular Arithmetic Misconceptions
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Introduction to Number Theory

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Chris Pinner CW205 [email protected] Office Hours (tentative): MWF 10:30 or by appointment. Home-Page: http://www.math.ksu.edu/~pinner/math506

Updates: 3/15/20 See Canvas for future homeworks & updates!

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100 Number Theory Problems (With Solutions)

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Crated on June, 2011. Problems are taken from IMO, IMO Shortlist/Longlist, and some other famous math competitions.

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Basic Number Theory-1

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  • Basics of Number Theory
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  • Program to find LCM of two numbers
  • Program for factorial of a number
  • Efficient program to print all prime factors of a given number
  • Binomial Coefficient | DP-9
  • Program for nth Catalan Number
  • Euclid's lemma
  • Euclidean algorithms (Basic and Extended)
  • Modular Arithmetic
  • Modular Addition
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  • Euler's Totient function for all numbers smaller than or equal to n
  • Modular Exponentiation (Power in Modular Arithmetic)
  • Program to find remainder without using modulo or % operator
  • Modular multiplicative inverse
  • Multiplicative order
  • Compute nCr%p using Lucas Theorem
  • Compute nCr%p using Fermat Little Theorem
  • Introduction to Chinese Remainder Theorem
  • Implementation of Chinese Remainder theorem (Inverse Modulo based implementation)
  • Find Square Root under Modulo p | Set 1 (When p is in form of 4*i + 3)
  • Find Square Root under Modulo p | Set 2 (Shanks Tonelli algorithm)
  • Cyclic Redundancy Check and Modulo-2 Division
  • Primitive root of a prime number n modulo n
  • Euler's criterion (Check if square root under modulo p exists)
  • Using Chinese Remainder Theorem to Combine Modular equations
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  • Wilson's Theorem
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  • Introduction to Primality Test and School Method
  • Fermat Method of Primality Test
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  • Legendre's formula (Given p and n, find the largest x such that p^x divides n!)
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Miscellaneous Practice Problems

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Number Theory for DSA & Competitive Programming

  • What is Number Theory?
Number theory is a branch of pure mathematics that deals with the properties and relationships of numbers, particularly integers. It explores the fundamental nature of numbers and their mathematical structures. Number theory has been studied for centuries and has deep connections with various areas of mathematics, including algebra, analysis, and geometry .

Table of Content

  • Practice Problems on Number Theory
  • Miscellaneous Practice Problems on Number Theory

Basics of Number Theory :

  • Find the GCD of two number
  • Find the LCM of two number
  • Calculate the Factorial of a number
  • Program to find Prime factors of a number
  • Find Binomial Coefficient C(n, k)
  • Program to find nth Catalan numbers  
  • Euclid’s Lemma
  • Basic and Extended Euclidean algorithms

Modular Arithmetic:

  • Euler’s Totient Function
  • Euler’s Totient function for all numbers smaller than or equal to n
  • Find the remainder without using the modulo operator
  • Compute nCr % p using Dynamic Programming Solution
  • Lucas Theorem
  • Compute nCr % p using Lucas Theorem
  • Compute nCr % p using Fermat Little Theorem
  • Fermat Little Theorem .
  • Chinese Remainder Theorem
  • Chinese Remainder Theorem using Inverse Modulo-based Implementation
  • Find primitive root of a prime number n modulo n
  • Euler’s criterion (Check if square root under modulo p exists)
  • Combine Modular equations using the Chinese Remainder Theorem
  • Wilson’s Theorem

Number Theory:

  • Primality Test to check if a number is prime or not
  • Primality Test to check if a number is prime or not using Fermat Method
  • Primality Test to check if a number is prime or not using Miller–Rabin
  • Primality Test to check if a number is prime or not using Solovay-Strassen
  • Legendre’s formula (Given p and n, find the largest x such that p^x divides n!)
  • Find if a number is Carmichael Numbers
  • Find all generators of cyclic additive group under modulo n
  • Fermat Number
  • Goldbach’s Conjecture
  • Pollard’s Rho Algorithm for Prime Factorization

Game Theory:

  • Minimax algorithm
  • Nim Game problem
  • Sprague – Grundy Theorem

Practice Problems on Number Theory:

Miscellaneous practice problems on number theory:.

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  1. PDF Introduction to Higher Mathematics Unit #4: Number Theory

    In this section we describe a few typical number theoretic problems, some of which we will eventually solve, some of which have known solutions too difficult for us to include, and some of which remain unsolved to this day. Sums of Squares I. Can the sum of two squares be a square? The answer is clearly "YES"; for example 32+4 = 52and 5 +12 = 132.

  2. PDF Practice Number Theory Problems

    Practice Number Theory Problems Problem 3-1. GCD Compute gcd(85; 289) using Euclid's extended algorithm. Then compute x and y such that 85x + 289y = gcd(85; 289). Recall Euclid's extended algorithm: = bq1 + r1 = r1q2 + r2 : : : rn 1 = rnqn+1 + rn+1: We stop when we reach a remainder of 0, that is, when rn+1 = 0. We obtain gcd(a; b) = rn.

  3. Number Theory Questions

    Solution: Prime factorisation of 8 = 2 × 2 × 2 Prime factorisation of 27 = 3 × 3 × 3 Prime factorisation of 72 = 2 × 2 × 2 × 3 × 3 LCM (8, 27, 72) = 2 × 2 × 2 × 3 × 3 × 3 = 216 Therefore, the LCM of 8, 27 and 72 is 216. 4. Write the first 10 non-zero multiples of 7. Solution:

  4. PDF Solutions to the Number Theory Problems

    Solutions to the Number Theory Problems 1: Show that p (2 + 3)n is odd for every positive integer n. p p n n p Solution: Notice that (2+ 3)n+(2 3)n = i 2n i 3 + i=0 p (2 3)n < 1, so pi P n ( 1)i 2n i bn=2c i 3 = 2 P i i=0 p (2 + 3)n j=0 which is an even number, and we have 0 < is odd. 2j n 2n 2j3j,

  5. Category:Introductory Number Theory Problems

    1. 1951 AHSME Problems/Problem 15. 1951 AHSME Problems/Problem 19. 1960 AHSME Problems/Problem 16. 1960 AHSME Problems/Problem 8. 1961 AHSME Problems/Problem 17. 1961 AHSME Problems/Problem 28. 1961 AHSME Problems/Problem 35. 1961 AHSME Problems/Problem 4.

  6. Practice Number Theory

    Introduction Perfecting Patterns Community Wiki Browse through thousands of Number Theory wikis written by our community of experts. Integers Even and Odd Numbers Divisibility Rules (2,3,5,7,11,13,17,19,...) Remainder Prime Numbers Infinitely Many Primes Distribution of Primes Mersenne Prime Prime Factorization Perfect Squares, Cubes, and Powers

  7. Elementary Number Theory

    Chapter 1: Preliminaries Section 1-1: Mathematical Induction Section 1-2: The Binomial Theorem Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise 5 Exercise 6 Exercise 7 Exercise 8 Exercise 9 Exercise 10 Exercise 11 Exercise 12 Exercise 13 Exercise 14 Chapter 2: Divisibility Theory in the Integers Section 2-1: Early Number Theory Section 2-2:

  8. PDF Here are some practice problems in number theory. They are, very roughly

    Here are some practice problems in number theory. They are, very roughly, in increasing order of difficulty. (a) Show that n7 − n is divisible by 42 for every positive integer n. (b) Show that every prime not equal to 2 or 5 divides infinitely many of the numbers 1, 11, 111, 1111, etc. Show that if p > 3 is a prime, then p2 ≡ 1 (mod 24).

  9. Exams

    Exams | Theory of Numbers | Mathematics | MIT OpenCourseWare Exams This section provides the two midterm exams and the final exam, along with a set of practice problems, exam guidelines, and solutions for each of the three exams.

  10. PDF 250 PROBLEMS IN ELEMENTARY NUMBER THEORY

    "250 Problems in Elementary Number Theory" presents problems and their solutions in five specific areas of this branch of mathe­ matics: divisibility of numbers, relatively prime numbers, arithmetic progressions, prime and composite numbers, and Diophantic equations. There is, in addition, a section of miscellaneous problems.

  11. PDF Problems in Elementary Number Theory

    AMM, Problem E2510, Saul Singer A 14. Let nbe an integer with n 2. Show that ndoes not divide 2n 1. A 15. Suppose that k 2 and n 1;n 2; ;n k 1 be natural numbers having the property n 2 j2n 1 1;n 3 j2n 2 1; ;n kj2n k 1 1;n 1 j2n k 1: Show that n

  12. PDF Intro to Number Theory: Solutions

    Intro to Number Theory: Solutions Dr. David M. Goulet November 14, 2007 Preliminaries Base 10 Arithmetic Problems • What is 7777 + 1 in base 8? Solution: In base 10, 7 + 1 = 8, but in base 7, 7 + 1 = 10. So 7777 + 1 = 7770 + 10 = 7700 + 100 = 7000 + 1000 = 10000. • In what base is 212 equal to 22510 ? Solution: call the base b.

  13. MATH506 Number Theory Homepage

    Exam Solutions Exam 1: Blank Solutions. Syllabus. Printable Syllabus Text: ... (McGraw-Hill ISBN -07232-571-2 is the same edition). Course Outline Number theory is essentially the study of the natural numbers 1,2,3,...and their properties. ... Its problems, often simple to state, have in many cases remained unsolved for centuries. ...

  14. PDF Open problems in number theory

    Examples : (3; 5) (5; 7) (11; 13) (17; 19) (29; 31) (41; 43) (1019; 1021) (2027; 2029) (3119; 3121) (4001; 4003) (5009; 5011)(6089; 6091) (7127; 7129) (8009; 8011) (9011; 9013) Landau's conjecture There are infinitely many primes of the form n2 + 1. Landau's conjecture There are infinitely many primes of the form n2 + 1. Examples : 2 = 12 + 1

  15. 100 Number Theory Problems (With Solutions)

    15. (IMO 1986, Day 1, Problem 1) Let d be any positive integer not equal to 2, 5 or 13. Show that one can find distinct a, b in the set {2, 5, 13, d} such that ab − 1 is not a perfect square. 16. (IMO 1988, Day 2, Problem 6) Let a and b be two positive integers 2 +b2 such that a · b + 1 divides a2 + b2 .

  16. Theory of Numbers, Exam 1 Practice

    Resource Type: Exams. pdf. Theory of Numbers, Exam 1 Practice. Download File. DOWNLOAD. This file contains information regarding Practice problems for Midterm1.

  17. Number theory

    Overview. Number theory is a broad topic, and may cover many diverse subtopics, such as: Modular arithmetic; Prime numbers; Some branches of number theory may only deal with a certain subset of the real numbers, such as integers, positive numbers, natural numbers, rational numbers, etc.Some algebraic topics such as Diophantine equations as well as some theorems concerning integer manipulation ...

  18. Introduction to Number Theory Textbook

    Introduction to Number Theory Textbook. Art of Problem Solving. Math texts, online classes, and more for students in grades 5-12. Engaging math books and online learning for students ages 6-13. Small live classes for advanced math and language arts learners in grades 2-12. ‚. MIT PRIMES/CrowdMath.

  19. Truth Tables Practice Problems With Answers

    There are eight (8) problems for you to work through in this section that will give you enough practice in constructing truth tables. Problem 1: Write the truth table for. Answer. Problem 2: Write the truth table for. Answer. Problem 3: Write the truth table for. Answer.

  20. Basic Number Theory-1 Practice Problems Math

    SOLVE NOW Prime Array ATTEMPTED BY: 1684 SUCCESS RATE: 78% LEVEL: Easy SOLVE NOW Alice and candies ATTEMPTED BY: 4631 SUCCESS RATE: 88% LEVEL: Medium SOLVE NOW Modulo Fermat's Theorem ATTEMPTED BY: 897 SUCCESS RATE: 36% LEVEL: Hard SOLVE NOW Archery ATTEMPTED BY: 3772 SUCCESS RATE: 70% LEVEL: Medium SOLVE NOW Mancunian and Pandigital Numbers

  21. PDF Solutions of the Algebraic Number Theory Exercises

    This article wants to be a solution book of Algebraic Number Theory. The solutions that would be presented are not o cial. Unless other-wise speci ed, all the references come from Algebraic Number Theory. Then \Theorem 1.7" will be the 7ththeorem in the rst part of the book.

  22. Number Theory for DSA & Competitive Programming

    Compute nCr % p using Dynamic Programming Solution; Lucas Theorem; Compute nCr % p using Lucas Theorem; Compute nCr % p using Fermat Little Theorem; Fermat Little Theorem. ... Miscellaneous Practice Problems on Number Theory: Problems. How to compute mod of a big number? BigInteger Class in Java: Modulo 10^9+7 (1000000007)

  23. Art of Problem Solving

    From 2017 to 2019 the test had 30 questions and lasted for 3 hours. In 2021-22, the contest was reduced to 12 questions. Since 2021-22, PRMO and RMO have been merged into IOQM. Each answer is a one or two digit positive integer. The test covers pre-college math topics, especially algebra, number theory, combinatorics, and geometry.