Mathematical Algorithms | GCD & LCM

Greatest Common Divisor (GCD):

The GCD of two numbers is defined as the largest integer that divides both integers without any remainder. It is also termed as the HCF (Highest Common Factor).

GCD of an array is the integer that divides all the elements of the array without leaving any remainder.

GCD of two numbers may be calculated using the following ways:

• Prime Factorization
• Divisions by Prime
• Euclidean Algorithm

Least Common Multiple (LCM):

The LCM of two numbers is defined as the smallest integer which is a multiple of both integers. 

LCM of an array is the smallest possible integer that is multiple of all the elements of the array.

Below are some process to find LCm of two numbers:

• Prime Factorization
• Divisions by Prime
• Using the relation between GCD and LCM

Relation between GCD and LCM:

Consider two integers a and b and their GCD be gcd(a, b) and LCM be lcm(a, b). Therefore, 

a * b = gcd(a, b) * lcm(a, b)

Some practice problems on GCD and LCM:

• Easy:
  1. GCD, LCM and Distributive Property
  2. Program to find GCD or HCF of two numbers
  3. Program to find LCM of two numbers
  4. Least Common Denominator (LCD)
  5. GCD of digits of a given number
  6. GCD of array
  7. LCM of array
  8. Find LCM of rational numbers
  9. GCD of two numbers when one of them can be very large
  10. GCD of elements in a given range
  11. LCM of First n Natural Numbers
  12. LCM of digits of a given number
  13. Program to find HCF iteratively
• Intermediate:
  1. Basic and Extended Euclidean algorithms
  2. Program to find GCD of floating point numbers
  3. HCF of array of fractions (or rational numbers)
  4. Pair with maximum GCD from two arrays
  5. LCM of factorial and its neighbors
  6. Largest subarray with GCD one
  7. Replace every matrix element with maximum of GCD of row or column
  8. Check if LCM of array elements is divisible by a prime number or not
  9. Print the kth common factor of two numbers
  10. Finding LCM of more than two (or array) numbers without using GCD
  11. Given GCD G and LCM L, find number of possible pairs (a, b)
  12. GCD of two numbers formed by n repeating x and y times
  13. Largest Subset with GCD 1
  14. Count pairs of natural numbers with GCD equal to given number
• Hard:
  1. Stein’s Algorithm for finding GCD
  2. Largest subsequence having GCD greater than 1
  3. Count number of pairs (A <= N, B <= N) such that gcd (A , B) is B
  4. Smallest number divisible by n and has at-least k trailing zeros
  5. Array with GCD of any of its subset belongs to the given array
  6. Find pair with maximum GCD in an array
  7. First N natural can be divided into two sets with given difference and co-prime sums
  8. Series with largest GCD and sum equals to n
  9. Maximum sum of distinct numbers with LCM as N
  10. Queries for GCD of all numbers of an array except elements in a given range
  11. Summation of GCD of all the pairs up to N
  12. Count number of subsets of a set with GCD equal to a given number
  13. Check if elements of array can be made equal by multiplying given prime numbers

Quick Links:

• Recent articles on GCD and LCM
• Practice problems on Math