The concept of lcm is extremely important in number theory and can be easily applied to problems related to work-time, speed-distance, and speed-time. It is mostly used in solving numerical problems such as finding the time taken for two lights to blink together or the ones based on circular tracks where we need to find the time at which the two runners meet. However, before we begin to learn how to find lcm, it is important to get familiar with a few common terms that will help us in calculating lcm. The first step of finding the lcm of two numbers is to have an idea of prime numbers. A prime number is a number that has no other divisor other than the number itself and 1. Numbers like 2,3,5,7,11 and so on are called as prime numbers as they only have 1 and their own number as their factors. However, if we take the number 6, we find that it has 2, 3 and 1 as its factors and is thus, not a prime number.
Lcm or the least common multiple of any two numbers x and y is the value of that smallest number that can divide both x and y. The lcm of any two numbers is only valid if the numbers are not equal to zero as in that case the results would be undefined because zero results in an undefined value on dividing it by some number x. Lcm can thus, be calculated for as many numbers as possible provided the numbers are not same or equal to zero. The easiest and the most common method that teaches us how to find lcm of two numbers is by using the method of prime factorization. This method makes note of the principle that every composite number is made up of prime numbers and can be factorized and written as the product of these primes. After factorizing the numbers, the product formed by multiplying only the highest powers of the all the prime numbers will give the value of lcm. This will become easier to understand when illustrated with a simple example. Take the numbers 8, 9 and 21 and factorize them. The number 8 can be written as 2X2X2 or 2^3 while 9 can be factorized as 3X3 or 3^2 and finally 21 can be written as 7x3. Now, the highest power of each of the three prime numbers 2, 3 and 7 is 2^3, 3^2 and 7^1. They are further multiplied to give the result as:
LCM of 21, 8 and 9= 2^3 X 3^2 X 7^1 = 8 X9 X7 = 504;
Among the various advantages this method provides, the most useful of them is its extremely fast nature of calculations for two and three digit numbers. It is also helpful in solving questions based on pipes and cisterns and work-time applications in very less duration. Its only small disadvantage is that the calculations become very complex as the number becomes bigger.
As discussed above, prime numbers are those numbers which do not have a divisor other than the number itself. Thus, factorizing a prime number is of no use as it will give us the same prime number. Also, no two prime numbers have a common factor and hence, its lcm is calculated by simply calculating the product of the two prime numbers. Take for example the numbers 5 and 11. Both of them are prime with no common factor other than 1 and so their lcm will be their product, that is, 55.
Similarly, before learning how to find lcm of coprime numbers, it is important to know the definition of coprime numbers. Numbers which do not have a common factor are known as coprime numbers and based on this definition, every prime number is a coprime number, but every coprime number is not prime. In fact, coprime numbers are either prime or composite numbers. Since two coprime numbers will never have any common factor, therefore, its lcm can also be found by simply finding the product of the numbers. For example, the composite numbers 15 and 14 are coprime as their factors 5X3 and 7X2 are not common and so their lcm will be the product of these two numbers which is 210. The method can be applied for any number of prime and composite numbers.
The most primitive or the earliest method known for calculating the lcm of two numbers is perhaps the Brute force method. This method is best used when the numbers are small and should be avoided in case of large numbers as the calculations will be very large. Here, instead of listing the factors we are supposed to list down the multiples of the numbers and then find out the smallest common multiple of both the numbers which will give the value of lcm. For example, if we take the numbers 6 and 4 and start writing their multiples, we will find that 12 is the least multiple that is divided by both 6 and 4 and thus, 12 is their lcm. Learning how to find lcm can be extremely easy if it is followed step by step.
Learning how to find lcm using the division method is very easy and most commonly taught in schools at a very young age. The very first step in the method is to write down all the numbers whose lcm has to be found in a single horizontal line that is divided into two parts. The divisor for this division method will only be a prime number that has to be written on the left side of the horizontal line. Divide the written numbers by a suitable divisor that can divide a few numbers. The quotient of the divisor is written in the next row of the number. All those numbers which were not divided by the common divisor are brought down as it is to next row. This process is repeated until the last row has only coprime numbers left. This coprime number is multiplied with the prime divisors of the numbers to give the value of lcm. Learning how to find lcm using the long division method is very useful as it can even be applied for extremely large numbers or when there are too many numbers whose lcm has to be determined.
While we have learnt how to find lcm using so many methods, it is also necessary to have some idea about finding the gcd or hcf of any pair of numbers. Hcf means the greatest common divisor that divides all the given number without leaving a possible remainder. There are two more names for hcf which are gcd and gcm. Finding hcf of two numbers is even simpler than knowing how to find lcm and thus, it can be calculated very easily. The most common method of finding hcf of two numbers is using prime factorization. Just as it is done in lcm, the numbers here are also written as the product of primes and the common primes are highlighted in the two numbers. This product of the common primes gives the value of hcf. For example, the numbers 4 and 6 can be written as the product of their primes as 2X2 and 2X3 respectively. The only common prime in both the numbers is 2 which is, in this case, their hcf. Just like lcm, hcf can also be found for more than two number provided the numbers are not equal to zero.
Hcf for small numbers can even be written in the form of a Venn diagram. The prime factors of the numbers whose hcf has to be calculated can be written in the sets A, B, C and so on. The common prime numbers for all the sets can be written in the subset that covers overlapped area. The prime numbers which are common to all the sets are listed below and their product gives the value of the hcf of those numbers. Hcf also has many mathematical applications especially in areas where we are required to deal with questions such as finding the largest tile that can applied on wall or the maximum capacity two drums can hold or in questions where we find the maximum persons among which a certain item needs to be distributed. Many remainder based questions can only be solved if we know the concept of hcf and lcm properly. Knowing how to find lcm and hcf is very simple if the above steps are followed and learning them can help us solve many mathematical problems that arise in our day to day life.