In mathematics computation of the least common multiple and greatest common divisors of two or more numbers. LCM is the smallest integer which is a multiple of two or more numbers. For example, LCM of 4 and 6 is 12, and LCM of 10 and 15 is 30. As with the greatest common divisors, there are many methods for computing the least common multiples also. One method is to factor both numbers into their primes. The LCM is the product of all primes that are common to all numbers. In this topic, we will discuss the concept of least common multiple and LCM formula with examples. Let us learn it!
LCM Formula
What is LCM?
The Least Common Multiple i.e. LCM of two integers a and b is that smallest positive integer which is divisible by both a and b. Thus the smallest positive number is a multiple of two or more numbers.
For example, to calculate lcm of (40, 45), we will find factors of 40 and 45, getting
40 is expressed as 2× 2× 2× 5
45 is expressed as 3× 3 × 5
The prime factors common to one or the other are 2, 2, 2, 3, 3, 5.
Thus the least common multiple will be 2× 2× 2× 3× 3× 5 = 360.
To find out LCM using prime factorization method:
Step 1: Show each number as a product of their prime factors.
Step 2: LCM will be the product of the highest powers of all prime factors.
To find out the LCM using division Method:
Step 1: First, we need to write the given numbers in a horizontal line separated by commas.
Step 2: Then, we need to divide all the given numbers by the smallest prime number.
Step 3: We now need to write the quotients and undivided numbers in a new line below the previous one.
Step 4: Repeat this process until we find a stage where no prime factor is common.
Step 5: LCM will be the product of all the divisors and the numbers in the last line.
L.C.M formula for any two numbers:
1) For two given numbers if we know their greatest common divisor i.e. GCD, then LCM can be calculated easily with the help of given formula:
LCM =\( \frac{a × b}{(gcd)(a,b)} \)
2) To get the LCM of two Fractions, then first we need to compute the LCM of Numerators and HCF of the Denominators. Further, both these results will be expressed as a fraction. Thus,
LCM = \(\frac{L.C.M\;of\;Numerator}{H.C.F\;of\;Denominator}\)
Solved Examples
Q.1: Find out the LCM of 8 and 14.
Solution:
Step 1: First write down each number as a product of prime factors.
8 = 2× 2× 2 = 2³
14 = 2× 7
Step 2: Product of highest powers of all prime factors.
Here the prime factors are 2 and 7
The highest power of 2 here = 2³
The highest power of 7 here = 7
Hence LCM = 2³× 7 = 56
Q.2: If two numbers 12 and 30 are given. HCF of these two is 6 then find their LCM.
Solution: We will use the simple formula of LCM and GCD.
a = 12
b =30
gcd =6
Thus
LCM = \(\frac{a\times b}{gcd\left(a,b\right)} \)
LCM = \(\frac {12 \times 30 } {6} \)
= 60
