Distributive Property of GCD and LCM

Distributive Property of the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) shows how GCD and LCM interact with each other when you have multiple numbers.

The distributive properties of the Greatest Common Divisor (GCD) and the Lowest Common Multiple (LCM) can be expressed in the following theorems for integers a, b, c ∈ Z (Z represents the set of all integers).

Theorem 1: LCM Distributive over GCD

This theorem states that when calculating the LCM of a number a with the GCD of two other numbers b and c, the result is equal to the GCD of the LCM of a with b and the LCM of a with c. Essentially, it shows how LCM interacts with the GCD operation.

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

Theorem 2: GCD Distributive over LCM

This theorem states that when calculating the GCD of a number a with the LCM of two other numbers b and c, the result is equal to the LCM of the GCD of a with b and the GCD of a with c.

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

Proof of Distributive Property of GCD and LCM

In this section, we will demonstrate the distributive properties of the Greatest Common Divisor (GCD) and the Lowest Common Multiple (LCM) through two key proofs:

Proof of Distributive Property of LCM over GCD

Consider a prime divisor p_s and its exponents s_a, s_b, and s_c in the numbers a, b, and c.

Let: x = LCM (a, GCD (b, c))

From the prime decomposition, the exponent of p_s in x is given by: max (s_a​, min (s_y, s_c​))

Using the property that max distributes over min, we can rewrite this as:

max (s_a​, min (s_b​, s_c​)) = min (max (s_a​, s_b​), max (s_a​, s_c​))

Thus, it follows that:

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

Let's consider an example for the same.

• Let a = 4, b = 6, and c = 8.
• Calculate LCM (4, GCD (6, 8)).
  • GCD (6, 8) = 2
  • LCM (4, 2) = 4
• Calculate GCD (LCM (4, 6), LCM (4, 8))
  • LCM (4, 6) = 12 and LCM (4, 8)=8
  • GCD (12, 8) = 4

Both sides equal 4, confirming the theorem.

Proof of Distributive Property of GCD over LCM

Similarly, consider the same prime divisor p_s​.

Let: y = GCD (a, LCM (b, c))

From the prime decomposition, the exponent of p_s​ in y is: min (s_a, max (s_b, s_c))

Using the property that min distributes over max, we can express this as:

min (s_a, max (s_b, s_c)) = max (min (s_a, s_b), min (s_a, s_c))

This shows that:

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

Let's consider an example for the same.

• Let a = 10, b = 15.
• Calculate GCD (10, LCM (15, 20))
  • LCM (15, 20) = 60
  • GCD (10, 60) = 10
• Calculate LCM (GCD (10, 15), GCD (10, 20))
  • GCD (10, 15)=5
  • LCM (5, 10)=10

Both sides equal 10, confirming this theorem as well.

Read More,

• Distributive Property
• HCF and LCM
• GCD Practice Problems with Solutions
• Short Tricks to Find GCD
• HCF and LCM of Polynomials
• Quiz about GCD