formula to find lcm
There are different ways to find the LCM (Least Common Multiple) of two or more numbers, but one common method is to use the prime factorization of each number. Here is a formula that involves the greatest common divisor (GCD) and the product of the numbers:
LCM(a,b) = |a*b| / GCD(a,b)
where |a*b| denotes the absolute value of the product of a and b, and GCD(a,b) denotes the greatest common divisor of a and b. This formula can be extended to finding the LCM of more than two numbers by applying it successively to pairs of numbers. For example:
LCM(a,b,c) = LCM(LCM(a,b),c)
Using the prime factorization method, we can first list the prime factors of each number, then pair up all the primes with their maximum exponent and multiply them together, as shown in the following example:
Find LCM(12, 30, 40)
Prime factorization:
12 = 2^2 * 3
30 = 2 * 3 * 5
40 = 2^3 * 5
Pair up the primes:
2^3 * 3 * 5 = 120
So, LCM(12, 30, 40) = 120
Note that the LCM is always a multiple of the given numbers, and it is the smallest positive integer that is divisible by all of them.
To find the Least Common Multiple (LCM) of two or more numbers, you can use the following steps:
Step 1: List the prime factors of each number.
Step 2: Write down all the prime factors with the highest exponent for each factor.
Step 3: Multiply all the prime factors together to get the LCM.
Let's demonstrate with an example:
Example: Find the LCM of 12 and 18.
Step 1: List the prime factors of 12 and 18.
- Prime factors of 12: 2, 2, 3
- Prime factors of 18: 2, 3, 3
Step 2: Write down all the prime factors with the highest exponent for each factor.
- Prime factors: 2 (twice), 3 (twice)
Step 3: Multiply all the prime factors together to get the LCM.
- LCM = 2 x 2 x 3 x 3 = 36
Therefore, the LCM of 12 and 18 is 36.