GCD (Greatest Common Divisor), also known as HCF (Highest Common Factor), is the largest positive integer that divides two or more numbers without leaving a remainder.
For example, the GCD of 20 and 30 is 10, as 10 is the largest number that divides both 20 and 30 evenly.
Here are some of the key properties of Greatest Common Divisor ( GCD ):
1. Commutative Property
The GCD is commutative, meaning the order of the numbers does not affect the result.
GCD (a, b) = GCD (b, a)
Example: GCD(48, 36) = 3 and GCD(36, 48) = 12
2. Associative Property
The GCD is associative, which means that when finding the GCD of multiple numbers, the grouping of numbers does not affect the result.
GCD (a, GCD (b, c)) = GCD (GCD (a, b), c)
Example: GCD(12, GCD(18, 24)) = GCD (12, 6) = 6 [As GCD (18, 24) = 6]
GCD(GCD(12, 18), 24) = GCD (6, 24) = 6. [As GCD (12, 18) = 6]
3. Distributive Property (over LCM)
The GCD is distributive over the LCM in the following way:
GCD (a, LCM (b, c)) = LCM (GCD (a, b), GCD (a, c))
Example: GCD (6, LCM (15, 10)) = LCM (GCD (6, 15), GCD (6, 10))
LCM (15, 10) = 30
- GCD (6, 15) = 3
- GCD (6, 10) = 2
- LCM (3, 2) = 6
GCD (6, 30) = 6
and LCM (3, 2) = 6
Thus, GCD (6, LCM (15, 10)) = LCM (GCD (6, 15), GCD (6, 10)) holds true.
4. Divisibility Property
The divisibility property of the Greatest Common Divisor (GCD) states that if d = GCD (a, b), then d divides both a and b without leaving any remainder.
d ∣ a and d ∣ b where d = GCD ( a, b )
Example: if d = GCD (8, 12) = 4, then 4 ∣ 8 and 4 ∣ 12.
5. GCD with Zero
The GCD of 0 and any non-zero integer (n) is n. However, the GCD of 0 and 0 is undefined, as there is no largest common divisor.
GCD ( 0, n ) = n but GCD ( 0, 0 ) = Undefined
GCD (12, 0) = 12 and GCD (0, 0) is undefined.
6. Multiplicative Property
The multiplicative property of GCD states that if a and b are co-prime, then:
GCD (a × b, c) = GCD (a, c) × GCD (b, c)
Example: GCD (5 × 3, 12) = GCD (5, 12) × GCD (3, 12) = 1 × 3 = 3.
- Verification: GCD (5 × 3, 12) = GCD (15 , 12) = 3
- Next Article - Euclid Division Lemma
- Read: Complete GCD Tutorial