The properties of LCM helps simplify complex calculations and provides insight into how numbers interact with each other. From its behavior with prime numbers to its relationship with zero, the properties of LCM form the foundation for solving a wide range of mathematical problems.
There are various properties of LCM including commutative property, associative property, distributive property, etc. which are very important to learn LCM.
Commutative Property
The order of the numbers does not affect the LCM.
LCM(a, b) = LCM(b, a)
Example: LCM(3, 13) = 39 and LCM(13, 3) = 13
Associative Property
The LCM is associative, which means that when finding the LCM of multiple numbers, the grouping of numbers does not affect the result.
LCM(a, b, c) = LCM (LCM (a, b), c) = LCM(a, LCM(b, c))
Example: LCM(10, 5, 7) = LCM (LCM (10, 5), 7) = LCM(10, LCM(5, 7))
Solution :
1. LCM(10, 5, 7) = 70 ..... (i)
2. Calculate LCM using (LCM(10, 5), 7):
LCM(10, 5) = 10, and then LCM(10, 7) = 70 ..... (ii)3. Calculate LCM using LCM(10, LCM(5, 7 ):
LCM(5, 7) = 35, and then LCM(10, 35) = 70 ..... (iii)From the result of (i), (ii), and (iii),
Thus, LCM(10, 5, 7) = 70 in all the cases , confirming the associative property.
Distributive Property
LCM supports the distributive property following this rule:
LCM(da, db, dc) = d × LCM(a, b, c)
Example: LCM(8, 12, 16)
Solution :
LCM(8, 12, 16) = 48
8 = 4 × 2
12 = 6 × 2
16 = 8 × 2Use Distributive Property:
LCM (8, 12, 16) = LCM (4 × 2, 6 × 2, 8 × 2)
LCM (4 × 2, 6 × 2, 8 × 2) = 2 × LCM (4, 6, 8)
48 = 2 × 24Thus, the distributive property is verified LCM (da, db, dc) = d × LCM (a, b, c).
Idempotent Property
The LCM of a number with itself is the number itself.
LCM(a, a) = a
Example: LCM(7, 7) = 7
LCM of Prime Numbers
The LCM of two distinct prime numbers is their product.
LCM(p, q) = p × q (where p and q are primes).
Example: LCM(5, 7) = 35
Note: If two numbers are coprime (i.e., their GCD is 1), then also the LCM is the product of the two numbers. Example: LCM(20, 27) = 20 × 27 = 540
LCM of a Number and its Multiple
The LCM of a two numbers when any one number is a multiple of the other number then the LCM is always equal to the larger number.
LCM(p, q) = q ( where q is multiple of p ).
Example: LCM(25, 75) = 75 as 75 is a multiple of 25.
Also Read: Interesting Facts about LCM
LCM and GCD Relationship
The product of the LCM and GCD of two numbers equals the product of the numbers themselves.
LCM(a, b) â‹… GCD(a, b) = a â‹… b
Example: LCM(12, 15) â‹… GCD(12, 15) = 12 â‹… 15
- LCM(12, 15) = 60, GCD(12, 15) = 3, and 60 â‹… 3 = 180 = 12 â‹… 15
LCM of 0
The LCM of any number and 0 is undefined, because every number is a multiple of 0, leading to an infinite set of multiples.
LCM(n, 0) = undefined
Example: LCM(0, 6) = undefined
LCM of a Number and 1
The LCM of any number and 1 is the number itself.
LCM(n, 1) = n
Example: LCM(12, 1) = 12.
- Next Article : Relationship between LCM and HCF