Mathematical Algorithms

  • Last Updated :22 Apr, 2022

Data Structure and Algorithms Course
Practice Problems on Mathematical Algorithms
Recent Articles on Mathematical Algorithms

Topic :

GCD and LCM :

  1. GCD and LCM
  2. LCM of array
  3. GCD of array
  4. Basic and Extended Euclidean algorithms
  5. Product of given N fractions in reduced form
  6. GCD of two numbers when one of them can be very large
  7. Stein’s Algorithm for finding GCD
  8. GCD, LCM and Distributive Property
  9. Replace every matrix element with maximum of GCD of row or column
  10. GCD of two numbers formed by n repeating x and y times
  11. Count number of pairs (A
  12. Array with GCD of any of its subset belongs to the given array
  13. First N natural can be divided into two sets with given difference and co-prime sums
  14. Minimum gcd operations to make all array elements one
  15. Program to find GCD of floating point numbers
  16. GCD of digits of a given number
  17. Series with largest GCD and sum equals to n
  18. Find pair with maximum GCD in an array
  19. GCD of elements in a given range
  20. Minimum operations to make GCD of array a multiple of k
  21. Largest Subset with GCD 1
  22. Queries for GCD of all numbers of an array except elements in a given range
  23. Summation of GCD of all the pairs up to N
  24. Largest subsequence having GCD greater than 1
  25. Largest subarray with GCD one

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Prime Factorization and Divisors :

  1. Prime factors
  2. Smith Numbers
  3. Sphenic Number
  4. Hoax Number
  5. k-th prime factor of a given number
  6. Pollard’s Rho Algorithm for Prime Factorization
  7. Prime Factorization using Sieve O(log n) for multiple queries
  8. Find politeness of a number
  9. Find sum of even factors of a number
  10. Find sum of odd factors of a number
  11. Find largest prime factor of a number
  12. Find minimum sum of factors of number
  13. Finding power of prime number p in n!
  14. Find all divisors of a natural number | Set 1
  15. Find all divisors of a natural number | Set 2
  16. Find all divisors of a natural number
  17. Find numbers with n-divisors in a given range
  18. Find minimum number to be divided to make a number a perfect square
  19. Sum of all proper divisors of a natural number
  20. Sum of all the factors of a number
  21. Sum of largest prime factor of each number less than equal to n
  22. Sum of all divisors from 1 to n
  23. Check for Amicable Pair
  24. Prime Factorization using Sieve O(log n) for multiple queries
  25. Prime factors of a big number

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Fibonacci Numbers :

  1. Fibonacci Numbers
  2. Interesting facts about Fibonacci numbers
  3. How to check if a given number is Fibonacci number?
  4. Zeckendorf’s Theorem (Non-Neighbouring Fibonacci Representation)
  5. Finding nth Fibonacci Number using Golden Ratio
  6. Matrix Exponentiation
  7. Fibonacci Coding
  8. n’th multiple of a number in Fibonacci Series
  9. GCD and Fibonacci Numbers
  10. Cassini’s Identity
  11. N-bonacci Numbers
  12. Space efficient iterative method to Fibonacci number
  13. The Magic of Fibonacci Numbers
  14. Program to print Fibonacci Triangle
  15. Factorial of each element in Fibonacci series
  16. Fibonomial coefficient and Fibonomial triangle
  17. Hosoya’s Triangle
  18. Prime numbers and Fibonacci
  19. Leonardo Number
  20. Fibonacci number in an array
  21. Fibonacci modulo p
  22. An efficient way to check whether n-th Fibonacci number is multiple of 10
  23. Find Index of given fibonacci number in constant time
  24. Tail Recursion for Fibonacci
  25. Large Fibonacci Numbers in Java
  26. Even Fibonacci Numbers Sum
  27. Nth Even Fibonacci Number
  28. Finding number of digits in n’th Fibonacci number
  29. Non Fibonacci Numbers
  30. Sum of Fibonacci Numbers
  31. Count ways to reach the n’th stair
  32. Count Possible Decodings of a given Digit Sequence
  33. Program to print first n Fibonacci Numbers | Set 1

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Modular Arithmetic :

  1. Modular Exponentiation (Power in Modular Arithmetic)
  2. Modular multiplicative inverse
  3. Modular Division
  4. Multiplicative order
  5. Find Square Root under Modulo p | Set 1 (When p is in form of 4*i + 3)
  6. Find Square Root under Modulo p | Set 2 (Shanks Tonelli algorithm)
  7. Euler’s criterion (Check if square root under modulo p exists)
  8. Multiply large integers under large modulo
  9. Find sum of modulo K of first N natural number
  10. How to compute mod of a big number?
  11. Modulo 10^9+7 (1000000007)
  12. How to avoid overflow in modular multiplication?
  13. Find (a^b)%m where ‘a’ is very large
  14. Find power of power under mod of a prime
  15. Number of solutions to Modular Equations
  16. Recursive sum of digits of a number formed by repeated appends
  17. Find value of y mod (2 raised to power x)
  18. Modular multiplicative inverse from 1 to n
  19. Find unit digit of x raised to power y
  20. Given two numbers a and b find all x such that a % x = b
  21. Exponential Squaring (Fast Modulo Multiplication)
  22. Subsequences of size three in an array whose sum is divisible by m
  23. Distributing M items in a circle of size N starting from K-th position
  24. Discrete logarithm (Find an integer k such that a^k is congruent modulo b)
  25. Finding ‘k’ such that its modulus with each array element is same
  26. Fibonacci modulo p
  27. Maximum subarray sum modulo m
  28. Trick for modular division ( (x1 * x2 …. xn) / b ) mod (m)
  29. Count number of solutions of x^2 = 1 (mod p) in given range
  30. Breaking an Integer to get Maximum Product
  31. Program to find remainder without using modulo or % operator

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Catalan Numbers :

  1. Catalan numbers
  2. Applications of Catalan Numbers
  3. Program for nth Catalan Number
  4. Dyck path
  5. Non-crossing lines to connect points in a circle
  6. Succinct Encoding of Binary Tree
  7. Find the number of valid parentheses expressions of given length

More >>

Euler Totient Function :

  1. Euler’s Totient Function
  2. Optimized Euler Totient Function for Multiple Evaluations
  3. Euler’s Totient function for all numbers smaller than or equal to n
  4. Primitive root of a prime number n modulo n
  5. Euler’s Four Square Identity

nCr Computations :

  1. Binomial Coefficient
  2. Compute nCr % p | Set 1 (Introduction and Dynamic Programming Solution)
  3. Compute nCr % p | Set 2 (Lucas Theorem)
  4. Compute nCr % p | Set 3 (Using Fermat Little Theorem)
  5. Program to calculate value of nCr
  6. Probability for three randomly chosen numbers to be in AP
  7. Rencontres Number (Counting partial derangements)
  8. Sum of squares of binomial coefficients
  9. Find sum of even index binomial coefficients
  10. Maximum binomial coefficient term value
  11. Program for Binomial Coefficients table
  12. Sum of Binomial coefficients
  13. Space and time efficient Binomial Coefficient
  14. Count ways to express even number ‘n’ as sum of even integers
  15. Maximum points of intersection n circles
  16. Horner’s Method for Polynomial Evaluation
  17. Print all possible combinations of r elements in a given array of size n
  18. Program to find the Volume of a Triangular Prism
  19. Significance of Pascal’s Identity
  20. Sum of all elements up to Nth row in a Pascal triangle

Chinese Remainder Theorem :

  1. Set 1 (Introduction)
  2. Set 2 (Inverse Modulo based Implementation)
  3. Cyclic Redundancy Check and Modulo-2 Division
  4. Using Chinese Remainder Theorem to Combine Modular equations

Factorial :

  1. Factorial
  2. Legendre’s formula (Given p and n, find the largest x such that p^x divides n!)
  3. Sum of divisors of factorial of a number
  4. Count Divisors of Factorial
  5. Compute n! under modulo p
  6. Double factorial
  7. Count trailing zeroes in factorial of a number
  8. Factorial of a large number
  9. Primorial of a number
  10. Find the first natural number whose factorial is divisible by x
  11. Count numbers formed by given two digit with sum having given digits
  12. Generate a list of n consecutive composite numbers (An interesting method)
  13. Expressing factorial n as sum of consecutive numbers
  14. Find maximum power of a number that divides a factorial
  15. Trailing number of 0s in product of two factorials
  16. Print factorials of a range in right aligned format
  17. GCD of factorials of two numbers
  18. Largest power of k in n! (factorial) where k may not be prime
  19. One line function for factorial of a number
  20. Find all factorial numbers less than or equal to n
  21. Find the last digit when factorial of A divides factorial of B
  22. An interesting solution to get all prime numbers smaller than n
  23. Calculating Factorials using Stirling Approximation
  24. Check if a number is a Krishnamurthy Number or not
  25. Find a range of composite numbers of given length
  26. Smallest number S such that N is a factor of S factorial or S!
  27. Maximum value of an integer for which factorial can be calculated on a machine
  28. Smallest number with at least n digits in factorial
  29. Last non-zero digit of a factorial
  30. Smallest number with at least n trailing zeroes in factorial
  31. Count natural numbers whose factorials are divisible by x but not y
  32. Count digits in a factorial | Set 1
  33. Count digits in a factorial | Set 2
  34. No of Factors of n!
  35. Count factorial numbers in a given range
  36. Program for factorial of a number

More >>

Prime numbers and Primality Tests :

  1. Prime Numbers
  2. Special prime numbers
  3. Prime numbers and Fibonacci
  4. Left-Truncatable Prime
  5. Mersenne Prime
  6. Super Prime
  7. Palindromic Primes
  8. Prime Triplet
  9. Hardy-Ramanujan Theorem
  10. Rosser’s Theorem
  11. Fermat’s little theorem
  12. Composite Number
  13. Euclid Euler Theorem
  14. Euclid’s Lemma
  15. Primality Test | Set 1 (Introduction and School Method)
  16. Primality Test | Set 2 (Fermat Method)
  17. Primality Test | Set 3 (Miller–Rabin)
  18. Primality Test | Set 4 (Solovay-Strassen)
  19. Primality Test | Set 5 (Using Lucas-Lehmer Series)
  20. Vantieghems Theorem for Primality Test
  21. AKS Primality Test
  22. Lucas Primality Test
  23. 2^x + 1 and Prime
  24. Recursive program for prime number
  25. Circular primes less than n

More >>

Set Theory :

  1. Minimize the absolute difference of sum of two subsets
  2. Sum of all subsets of a set formed by first n natural numbers
  3. Sum of average of all subsets
  4. Bell Numbers (Number of ways to Partition a Set)
  5. Power Set

Sieve Algorithms :

  1. Sieve of Eratosthenes
  2. Segmented Sieve
  3. Sieve of Atkin
  4. Sieve of Sundaram to print all primes smaller than n
  5. Sieve of Eratosthenes in 0(n) time complexity
  6. Arithmetic Number

More >>

Divisibility and Large Numbers :

  1. Check if a large number is divisible by 3 or not
  2. Number of digits to be removed to make a number divisible by 3
  3. Find whether a given integer is a power of 3 or not
  4. Check if a large number is divisible by 4 or not
  5. Count rotations divisible by 4
  6. Number of substrings divisible by 4 in a string of integers
  7. Check if a large number is divisible by 6 or not
  8. Prove that atleast one of three consecutive even numbers is divisible by 6
  9. Sum of all numbers divisible by 6 in a given range
  10. Number of substrings divisible by 6 in a string of integers
  11. Print digit’s position to be removed to make a number divisible by 6
  12. Check divisibility by 7
  13. To check whether a large number is divisible by 7
  14. Remainder with 7 for large numbers
  15. Count rotations divisible by 8
  16. Given a large number, check if a subsequence of digits is divisible by 8
  17. Check if a large number is divisible by 9 or not
  18. Decimal representation of given binary string is divisible by 10 or not
  19. Check if a large number is divisible by 11 or not
  20. Program to find remainder when large number is divided by 11
  21. Divisibility by 12 for a large number
  22. Check if a large number is divisible by 13 or not
  23. Check if a large number is divisibility by 15
  24. Check if a large number is divisible by 20
  25. Number is divisible by 29 or not

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Series :

  1. Juggler Sequence
  2. Padovan Sequence
  3. Aliquot Sequence
  4. Moser-de Bruijn Sequence
  5. Stern-Brocot Sequence
  6. Newman-Conway Sequence
  7. Sylvester’s sequence
  8. Recaman’s sequence
  9. Abundant Number
  10. Hexagonal Number
  11. Emirp numbers
  12. Nicomachus’s Theorem (Sum of k-th group of odd positive numbers)
  13. Sum of pairwise products
  14. Squared triangular number (Sum of cubes)
  15. Square pyramidal number (Sum of Squares)
  16. Program to print the sum of the given nth term
  17. Sum of series with alternate signed squares of AP
  18. Program for sum of cos(x) series
  19. Sum of range in a series of first odd then even natural numbers
  20. Sum of the sequence 2, 22, 222, ………
  21. Sum of the series 5+55+555+.. up to n terms
  22. Sum of series 1^2 + 3^2 + 5^2 + . . . + (2*n – 1)^2
  23. Sum of series 2/3 – 4/5 + 6/7 – 8/9 + ——- upto n terms
  24. Sum of the series 0.6, 0.06, 0.006, 0.0006, …to n terms
  25. n-th term in series 2, 12, 36, 80, 150….
  26. Program to print tetrahedral numbers upto Nth term

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Number Digits :

  1. n-th number whose sum of digits is ten
  2. Minimum digits to remove to make a number Perfect Square
  3. Count digits in given number N which divide N
  4. Count digit groupings of a number with given constraints
  5. Print first k digits of 1/n where n is a positive integer
  6. Program to check if a given number is Lucky
  7. Check if a given number can be represented in given a no. of digits in any base
  8. Find element using minimum segments in Seven Segment Display
  9. Find nth term of the Dragon Curve Sequence
  10. Find the Largest Cube formed by Deleting minimum Digits from a number
  11. Find next greater number with same set of digits
  12. Find the Number which contain the digit d
  13. Find nth number that contains the digit k or divisible by k
  14. Find N integers with given difference between product and sum
  15. Reverse a number using stack
  16. Check if a number is jumbled or not
  17. Number of digits in the product of two numbers
  18. Form the smallest number using at most one swap operation
  19. Difference between sums of odd and even digits
  20. Numbers having difference with digit sum more than s
  21. Count n digit numbers not having a particular digit
  22. Program to check Plus Perfect Number
  23. Total numbers with no repeated digits in a range
  24. K-th digit in ‘a’ raised to power ‘b’
  25. Possible to make a divisible by 3 number using all digits in an array

More >>

Triangles :

  1. Time required to meet in equilateral triangle
  2. Trinomial Triangle
  3. Leibniz harmonic triangle
  4. Hosoya’s Triangle
  5. Number of triangles after N moves
  6. Find Perimeter of a triangle
  7. Check whether right angled triangle is valid or not for large sides
  8. Maximum height of triangular arrangement of array values
  9. Find other two sides of a right angle triangle
  10. Find coordinates of the triangle given midpoint of each side
  11. Number of possible Triangles in a Cartesian coordinate system
  12. Triangular Numbers
  13. Pascal’s Triangle

Algebra :

  1. Find x and y satisfying ax + by = n
  2. Calculate the Discriminant Value
  3. Program for dot product and cross product of two vectors
  4. Iterated Logarithm log*(n)
  5. Program to find correlation coefficient
  6. Program for Muller Method
  7. Number of sextuplets (or six values) that satisfy an equation
  8. Complete the sequence generated by a polynomial
  9. Find the minimum value of m that satisfies ax + by = m and all values after m also satisfy
  10. Roots of Unity
  11. Number of non-negative integral solutions of a + b + c = n
  12. Roots of quadratic equation
  13. Find smallest values of x and y such that ax – by = 0
  14. Generate Pythagorean Triplets
  15. Square root of an integer
  16. Find number of solutions of a linear equation of n variables
  17. Write an iterative O(Log y) function for pow(x, y)
  18. Program to add two polynomials
  19. Multiply two polynomials
  20. Count Distinct Non-Negative Integer Pairs (x, y) that Satisfy the Inequality x*x + y*y
  21. Fast method to calculate inverse square root of a floating point number in IEEE 754 format
  22. Efficient program to calculate e^x

Number System :

  1. Exponential notation of a decimal number
  2. Check if a number is power of k using base changing method
  3. Check if number is palindrome or not in Octal
  4. Check if a number N starts with 1 in b-base
  5. Convert a binary number to hexadecimal number
  6. Program for decimal to hexadecimal conversion
  7. Converting a Real Number (between 0 and 1) to Binary String
  8. Count of Binary Digit numbers smaller than N
  9. Write a program to add two numbers in base 14
  10. Convert from any base to decimal and vice versa
  11. Decimal to binary conversion without using arithmetic operators

Misc :

  1. Tau – A Mathematical Constant
  2. Interquartile Range (IQR)
  3. Simulated Annealing
  4. Break the number into three parts
  5. Pseudo Random Number Generator (PRNG)
  6. Square root of a number using log
  7. Find ways an Integer can be expressed as sum of n-th power of unique natural numbers
  8. N’th palindrome of K digits
  9. N-th root of a number
  10. Fast Fourier Transformation for poynomial multiplication
  11. Find Harmonic mean using Arithmetic mean and Geometric mean
  12. Number of visible boxes after putting one inside another
  13. Generate a pythagoras triplet from a single integer
  14. Double Base Palindrome
  15. Program for Derivative of a Polynomial
  16. Sgn value of a polynomial
  17. Represent a number as sum of minimum possible psuedobinary numbers
  18. Program to print table of a number
  19. Compute average of two numbers without overflow
  20. Round-off a number to a given number of significant digits
  21. Convert a number m to n using minimum number of given operations
  22. Count numbers which can be constructed using two numbers
  23. Find Cube Pairs (A n^(2/3) Solution)
  24. Find the minimum difference between Shifted tables of two numbers
  25. Check if a number is a power of another number
  26. Check perfect square using addition/subtraction
  27. Number of perfect squares between two given numbers
  28. Count Derangements (Permutation such that no element appears in its original position)
  29. Print squares of first n natural numbers without using *, / and –
  30. Program to evaluate simple expressions
  31. Generate all unique partitions of an integer
  32. Random number generator in arbitrary probability distribution fashion
  33. Program to convert a given number to words
  34. Make a fair coin from a biased coin
  35. Generate integer from 1 to 7 with equal probability
  36. Print all combinations of balanced parentheses
  37. Print all combinations of points that can compose a given number
  38. Implement *, – and / operations using only + arithmetic operator
  39. Program to calculate area of an Circle inscribed in a Square
  40. Program to find the Area of Pentagon
  41. Program to find the Area and Volume of Icosahedron

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