Greatest Common Divisor (GCD)
2026-02-28 17:57 Diff
https://brightchamps.com/en-us/math/numbers/greatest-common-divisor

The greatest common divisor (GCD) obeys certain mathematical properties. Here are a few important ones:

Commutative Property

Since the order of the numbers does not change the final result, GCD is commutative, i.e., GCD \( (a, b) = \) GCD\( (b, a)\)

For example: GCD\((12, 18) = \).GCD\((18, 12) = 6\)

Associative Property

The GCD is associative because the grouping of numbers does not change the result, i.e., GCD (\(a, \)GCD\((b, c)) =\) GCD(GCD\( (a, b), c)\)

For example,

  • First Grouping: GCD(\(8,\) GCD \((12,16))\)

GCD\((12, 16) = 4\)

GCD \((8, 4) = 4\)

So, GCD(8, GCD \((12, 16)) = 4\)

  •  Second Grouping: GCD(GCD\((8, 12), 16)\)

GCD\((8, 12) = 4\)

GCD\((4, 16) = 4\)

So, GCD(GCD\((8, 12), 16) = 4\)

GCD\( (8,\) GCD\((12, 16)) =\) GCD(GCD\((8, 12), 16)\)

Hence, GCD is associative.

Distributive Property (Over LCM)

The GCD is distributive over the Least Common Multiple (LCM). 

GCD\( (a, \)LCM\( (b, c)) = \)LCM (GCD \((a, b)\), GCD\((a, c)).\)

For example: GCD\( (8, \)LCM\( (12, 18)) = \)LCM (GCD \((8, 12),\) GCD \((8, 18))\)

Let’s look at the steps involved:

  • LCM \((12, 18) = 36 \)
     
  • GCD\( (8, 12) = 4\)
     
  • GCD\( (8, 18) = 2\)
     
  • LCM \((4, 2) = 4\)
     
  • GCD\( (8, 36) = 4\)
     

Therefore, 

GCD \((8,\) LCM\( (12, 18))\) = LCM (GCD\( (8, 12)\), GCD \((8, 18)) \)holds true.

Divisibility Property

According to the divisibility property, if d is given as GCD \((a, b),\) then it means d divides both a and b evenly. 
\(d ∣ a \) and \(d ∣ b\) where d = GCD (\(a, b)\).

Example: If d = GCD \((18, 24) = 6\), then \(6 ∣ 18\) and \( 6 ∣ 24\). 

GCD with Zero

For any non-zero integer n, the GCD of 0 and n is n. However, the GCD of 0 and 0 is undefined because there is no greatest common divisor in this case.

GCD\((0, n) = n\), but GCD\((0, 0) =\) undefined.
 

Example: GCD\((12, 0) = 12\), but GCD\((0, 0)\) is undefined.

Multiplicative Property

According to the multiplicative property of GCD, if a and b are co-prime, then:

GCD\( (a × b, c) =\) GCD\( (a, c) ×\) GCD \((b, c). \)

For example:  Consider a = 7, b = 5, and c = 20.

Since 7 and 5 are co-prime;

(GCD \((7,5) = 1)\), GCD \((7 × 5, 20) =\) GCD \((7,20) ×\) GCD \((5,20)\) \(= 1 × 5 = 5 \)

Hence, the property holds true.