Interesting Facts about GCD (original) (raw)

Last Updated : 23 Jul, 2025

Greatest Common Divisor (GCD) is the largest number, which divides all the given integers exactly without leaving any remainder. For two integers (a and b) the greatest common divisor of a and b can be denoted as **GCD(a, b).

Some Amazing facts related to GCD are:

• The GCD of any number and zero is the number itself.
• The GCD of any two prime numbers is always 1, as they have no common divisors other than 1.
• GCD follows distributive property. GCD(n x a, n x b) = n x GCD(a, b) where a, n and b are greater than 0. For example GCD(15, 21) = 3 x GCD(5, 7) = 3 x 1 = 3.
• If two numbers have a GCD of 1, they are called **coprime or **relatively prime.
• If a number divides another, then the number itself is the GCD of the two numbers. For example, the GCD of 18 and 6 is 6.
• GCD is related to LCM by the formula: **GCD(a, b)×LCM(a, b) = a × b.
• The GCD of two consecutive numbers in the **Fibonacci sequence is always 1. For example, for Fibonacci numbers 21 and 34, GCD(21, 34) = 1.
• To find the GCD of multiple numbers, use GCD(a, b, c) = GCD(GCD(a, b), c).
• The GCD of any number with itself is that number, so GCD(a, a) = a.
• The GCD is based on absolute values, so GCD(−a, b) = GCD(a, b).
• The GCD of and b is equal to the GCD of b and a%b. This is the basis for the popular Euclid Algorithm to find GCD.
• GCD of two polynomials is the polynomial of the highest degree that divides both without a remainder.
• If two numbers are powers of the same base, say a = xm and b = xn, then their GCD is xmin⁡(m,n). For example, GCD(25, 23) = 23.
• GCD of a and b (where both a and b are non-zero), can also be defined as the smallest positive integer d which can be a solution/which can be expressed as a linear combination of a and b in the form d = a*p + b*q, where both p and q are integers
• **Extended Euclidean Algorithm (based on the previous fact) not only finds the GCD of two numbers but also provides **Bezout's coefficients, which are integers x and y such that a × x + b × y = GCD(a, b).
• For any two consecutive integers, the GCD is always 1. This is because consecutive numbers are always relatively prime. For example, GCD(100, 101) = 1.
• GCD and LCM distribute over each other.
  GCD(a, LCM(b, c)) = GCD(LCM(a, b), LCM(a, c)) and
  LCM(a, GCD(b, c)) = LCM(GCD(a, b), GCD(a, c))

**Read More,

• **HCF and LCM
• **Amazing Facts about Prime Numbers
• **Interesting facts about Fibonacci numbers
• **Interesting Facts about Infinity