gcd() in Python

Last Updated : 17 Mar, 2026

The Greatest Common Divisor (GCD) of two integers is the largest positive integer that divides both numbers without leaving a remainder. Python provides multiple ways to compute the GCD.

Example:

Python 
import math
a = 60
b = 48

print(math.gcd(a, b))

Output
12

Explanation: The common divisors of 60 and 48 are [1, 2, 3, 4, 6, 12] and the greatest one is 12.

Syntax

math.gcd(x, y)

Parameters: x, y Two integers (at least one of them must be non-zero).

Returns:

  • The largest integer that divides both numbers.
  • If one number is 0, the GCD is the absolute value of the non-zero number.

Using the math.gcd() Function

Python's math module provides a built-in gcd() function. It computes the GCD of two numbers easily.

Python 
import math

a = 60
b = 48
print(math.gcd(a, b))

Output
12

Explanation: The common divisors of 60 and 48 are [1, 2, 3, 4, 6, 12]. The greatest one is 12.

Examples

Example 1: GCD of a number and 0

Python 
import math

print(math.gcd(0, 25))

Output
25

Explanation: When one number is 0, the gcd() returns the absolute value of the other number.

Example 2: GCD of two co-prime numbers

Python 
import math

print(math.gcd(17, 29))

Output
1

Explanation: 17 and 29 are co-prime (no common divisors except 1), so gcd() returns 1.

Iterative (Brute Force) Approach

This method checks all numbers from 1 to the smaller of the two integers and finds the largest number that divides both.

Python 
def gcd_iterative(a, b):
    a, b = abs(a), abs(b)  
    gcd = 1
    for i in range(1, min(a, b) + 1):
        if a % i == 0 and b % i == 0:
            gcd = i
    return gcd
print(gcd_iterative(60, 48))

Output
12

Explanation:

  • a, b = abs(a), abs(b): Converts both numbers to positive to handle negative inputs.
  • gcd = 1: Initializes the variable to store the greatest common divisor.
  • for i in range(1, min(a, b) + 1): Loops through all numbers from 1 to the smaller of a and b.
  • if a % i == 0 and b % i == 0: Checks if i divides both a and b without a remainder.
  • gcd = i: Updates gcd whenever a common divisor is found.

Euclidean Algorithm

The Euclidean algorithm is an efficient way to compute GCD using the property:

Python 
def gcd_euclidean(a, b):
    a, b = abs(a), abs(b)  
    while b != 0:
        a, b = b, a % b
    return a

print(gcd_euclidean(60, 48))

Output
12

Explanation:

  • a, b = abs(a), abs(b): Converts both numbers to positive to handle negative inputs.
  • while b != 0: Loops until b becomes 0.
  • a, b = b, a % b: Updates a to b and b to the remainder of a ÷ b. This is the key step of the Euclidean algorithm.
  • return a: When b becomes 0, a contains the greatest common divisor.

GCD of Multiple Numbers Using functools.reduce()

To find the GCD of a list of numbers, math.gcd() can be applied one by one to all the numbers using functools.reduce(). This gives the greatest common divisor of the entire list.

Python 
import math
from functools import reduce

nums = [48, 64, 80]
res = reduce(math.gcd, nums)
print(res)

Output
16

Explanation:

  • reduce(math.gcd, nums) applies math.gcd() to the first two elements, then applies it to the result and the next element, and so on.
  • For nums = [48, 64, 80], first gcd(48, 64) = 16, then gcd(16, 80) = 16.
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