# Mathematical Algorithms

**Data Structure and Algorithms Course**

**Practice Problems on Mathematical Algorithms**

**Recent Articles on Mathematical Algorithms**

**Topic :**

- GCD and LCM
- LCM of array
- GCD of array
- Basic and Extended Euclidean algorithms
- Product of given N fractions in reduced form
- GCD of two numbers when one of them can be very large
- Stein’s Algorithm for finding GCD
- GCD, LCM and Distributive Property
- Replace every matrix element with maximum of GCD of row or column
- GCD of two numbers formed by n repeating x and y times
- Count number of pairs (A <= N, B <= N) such that gcd (A , B) is B
- Array with GCD of any of its subset belongs to the given array
- First N natural can be divided into two sets with given difference and co-prime sums
- Minimum gcd operations to make all array elements one
- Program to find GCD of floating point numbers
- GCD of digits of a given number
- Series with largest GCD and sum equals to n
- Find pair with maximum GCD in an array
- GCD of elements in a given range
- Minimum operations to make GCD of array a multiple of k
- Largest Subset with GCD 1
- Queries for GCD of all numbers of an array except elements in a given range
- Summation of GCD of all the pairs up to N
- Largest subsequence having GCD greater than 1
- Largest subarray with GCD one

**Prime Factorization and Divisors :**

- Prime factors
- Smith Numbers
- Sphenic Number
- Hoax Number
- k-th prime factor of a given number
- Pollard’s Rho Algorithm for Prime Factorization
- Prime Factorization using Sieve O(log n) for multiple queries
- Find politeness of a number
- Find sum of even factors of a number
- Find sum of odd factors of a number
- Find largest prime factor of a number
- Find minimum sum of factors of number
- Finding power of prime number p in n!
- Find all divisors of a natural number | Set 1
- Find all divisors of a natural number | Set 2
- Find all divisors of a natural number
- Find numbers with n-divisors in a given range
- Find minimum number to be divided to make a number a perfect square
- Sum of all proper divisors of a natural number
- Sum of all the factors of a number
- Sum of largest prime factor of each number less than equal to n
- Sum of all divisors from 1 to n
- Check for Amicable Pair
- Prime Factorization using Sieve O(log n) for multiple queries
- Prime factors of a big number

- Fibonacci Numbers
- Interesting facts about Fibonacci numbers
- How to check if a given number is Fibonacci number?
- Zeckendorf’s Theorem (Non-Neighbouring Fibonacci Representation)
- Finding nth Fibonacci Number using Golden Ratio
- Matrix Exponentiation
- Fibonacci Coding
- n’th multiple of a number in Fibonacci Series
- GCD and Fibonacci Numbers
- Cassini’s Identity
- N-bonacci Numbers
- Space efficient iterative method to Fibonacci number
- The Magic of Fibonacci Numbers
- Program to print Fibonacci Triangle
- Factorial of each element in Fibonacci series
- Fibonomial coefficient and Fibonomial triangle
- Hosoya’s Triangle
- Prime numbers and Fibonacci
- Leonardo Number
- Fibonacci number in an array
- Fibonacci modulo p
- An efficient way to check whether n-th Fibonacci number is multiple of 10
- Find Index of given fibonacci number in constant time
- Tail Recursion for Fibonacci
- Large Fibonacci Numbers in Java
- Even Fibonacci Numbers Sum
- Nth Even Fibonacci Number
- Finding number of digits in n’th Fibonacci number
- Non Fibonacci Numbers
- Sum of Fibonacci Numbers
- Count ways to reach the n’th stair
- Count Possible Decodings of a given Digit Sequence
- Program to print first n Fibonacci Numbers | Set 1

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

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

- Factorial
- Legendre’s formula (Given p and n, find the largest x such that p^x divides n!)
- Sum of divisors of factorial of a number
- Count Divisors of Factorial
- Compute n! under modulo p
- Double factorial
- Count trailing zeroes in factorial of a number
- Factorial of a large number
- Primorial of a number
- Find the first natural number whose factorial is divisible by x
- Count numbers formed by given two digit with sum having given digits
- Generate a list of n consecutive composite numbers (An interesting method)
- Expressing factorial n as sum of consecutive numbers
- Find maximum power of a number that divides a factorial
- Trailing number of 0s in product of two factorials
- Print factorials of a range in right aligned format
- GCD of factorials of two numbers
- Largest power of k in n! (factorial) where k may not be prime
- One line function for factorial of a number
- Find all factorial numbers less than or equal to n
- Find the last digit when factorial of A divides factorial of B
- An interesting solution to get all prime numbers smaller than n
- Calculating Factorials using Stirling Approximation
- Check if a number is a Krishnamurthy Number or not
- Find a range of composite numbers of given length
- Smallest number S such that N is a factor of S factorial or S!
- Maximum value of an integer for which factorial can be calculated on a machine
- Smallest number with at least n digits in factorial
- Last non-zero digit of a factorial
- Smallest number with at least n trailing zeroes in factorial
- Count natural numbers whose factorials are divisible by x but not y
- Count digits in a factorial | Set 1
- Count digits in a factorial | Set 2
- No of Factors of n!
- Count factorial numbers in a given range
- Program for factorial of a number

**Prime numbers and Primality Tests :**

- Prime Numbers
- Special prime numbers
- Prime numbers and Fibonacci
- Left-Truncatable Prime
- Mersenne Prime
- Super Prime
- Palindromic Primes
- Prime Triplet
- Hardy-Ramanujan Theorem
- Rosser’s Theorem
- Fermat’s little theorem
- Composite Number
- Euclid Euler Theorem
- Euclid’s Lemma
- Primality Test | Set 1 (Introduction and School Method)
- Primality Test | Set 2 (Fermat Method)
- Primality Test | Set 3 (Miller–Rabin)
- Primality Test | Set 4 (Solovay-Strassen)
- Primality Test | Set 5 (Using Lucas-Lehmer Series)
- Vantieghems Theorem for Primality Test
- AKS Primality Test
- Lucas Primality Test
- 2^x + 1 and Prime
- Recursive program for prime number
- Circular primes less than n

**Divisibility and Large Numbers :**

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

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

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

- Time required to meet in equilateral triangle
- Trinomial Triangle
- Leibniz harmonic triangle
- Hosoya’s Triangle
- Number of triangles after N moves
- Find Perimeter of a triangle
- Check whether right angled triangle is valid or not for large sides
- Maximum height of triangular arrangement of array values
- Find other two sides of a right angle triangle
- Find coordinates of the triangle given midpoint of each side
- Number of possible Triangles in a Cartesian coordinate system
- Triangular Numbers
- Pascal’s Triangle

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

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

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

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