LCM (Least Common Multiple) is the smallest positive integer that is divisible by each of the given numbers. Generally, we use prime factorization or division method to find LCM, but for large numbers, it can take a lot of time to find all the prime factors. In such cases, we can use the relationship between GCD and LCM to quickly find the LCM of the required numbers.
In this, we use the following formula:
LCM(a,b) = ∣a×b∣/GCD(a,b)
For example, find the LCM of 18 and 24.
Solution:
To find the GCD of 18 and 24, we use the Euclidean algorithm:
- 24 ÷ 18 = 1 (remainder 6)
- 18 ÷ 6 = 3 (remainder 0)
So, the GCD(18, 24) = 6.
Now, LCM(18,24) = (18 × 24)/6 = 432/6 = 72
The LCM of 18 and 24 is 72.
Other Tricks for LCM
Some other tips and tricks to find LCM are:
- If the given two numbers are co-prime (not having any prime factor common), LCM is the multiplication of the two numbers. For example LCM (6, 7) = 42.
- If one number is divisible by another number, the LCM is equal to the larger number. For example, LCM of 6 and 24 is 24.
- If two given numbers are different powers of a common number, simply the larger one is our result. For example, 32 and 64 are both powers of 2, so our result is 64.
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