HCF and LCM are calculated for a group of numbers. Such concepts are used everywhere, from sports to business. In this article, we look at the full forms, examples, prime factorization method, long division method, numerical problems, and formulas of HCF and LCM.
The full form of HCF and LCM:
HCF
The full form of HCF is the highest common factor – It is the largest possible number that can divide each of the numbers in a group without leaving a reminder. In other words, all the numbers in the group can be exactly divided by HCF. For example, the HCF of 12 and 6 is 6. The numbers 12 and 6 have common divisors 1,2,3 and 6. So, 6 will be the HCF.
LCM
The full form of LCM is the Least Common multiple – It is the least possible number that is a multiple of each of the numbers in the group. For example, the LCM of 8 and 12 is 48.
The multiples of 8 are: 8, 16, 32, 48, 56, 64, 72, 80…..
The multiples of 12 are: 12,24, 36, 48, 60, 72, 84, 96, 108, 120……
Though 48 and 72 are both common multiples of 8 and 12, the least common multiplier(LCM) is 48.
What if two numbers don’t have a common factor?
If two numbers don’t have a common divisor, as in the case of 3 (3*1) and 7 (7*1), the HCF will be one, as the number 1 can exactly divide any number in mathematics.
What if two numbers don’t have a common multiple?
This case is not possible. Let’s again take the example of 8 and 12. The product of 8*12 is 96, which is a multiple of both numbers. In this way the product of the numbers will always be a common multiple to each of the numbers. In the case of 8 and 12, 96 is not the least multiple, they have common multiples 48, 72, and 96. So, the LCM of 8 and 12 is 48 and not 96.
How to find HCF and LCM of two numbers
The HCF and LCM of two numbers can be calculated using the following two methods. In fact, we can calculate the HCF and LCM of multiple numbers using the methods below.
Prime Factorization method
In this method, we list out the primer factors of each of the numbers and accordingly calculate the greatest common divisor and the least common multiple.
Long division method
In this method, we divide the greater number by the smaller number to find out the HCF and LCM. This method is generally used when we have to find HCF and LCM of two numbers only.
HCF by prime factorization method
In the method, we factorize each of the numbers and find the common factors. We use the smallest possible prime factors to start with and then finally stop when the number cannot be factorized further.
Let us take the example of 48, 64 and 108
48 = 2 * 2 * 2 * 2 * 3
64 = 2 * 2 * 2 * 2 * 2 * 2
108 = 2 * 2 * 3 * 3 * 3
HCF = common factors = 2 * 2 = 4
Let us look at another example: HCF of two numbers 404 and 96
404 = 2 * 2 * 101 (It cannot be further factorized as 101 is a prime number)
96 = 2 * 2 * 2 * 2 * 2 * 3
HCF = common factor = 2 * 2 = 4
HCF by long division method
We will use the same example we used for prime factorization to demonstrate the long division method.
HCF of two numbers 404 and 96 by long division method.
- Divide the greater number with the smaller one.
- Divide the reminder by the last quotient
- Stop when the reminder becomes zero. (the divisor in this step is the HCF)
In the long division method, if the division does not result in a zero reminder, that means the two numbers don’t have a common factor and the HCF will 1 in that case. Example: HCF of 7 and 9 will be 1.
LCM by prime factorization
In this method, we list out all the prime factors of the numbers. Then we group the common prime factors and multiply them with the rest of the prime factors. Let us the example of the LCM of 8 and 12 to understand this further.
We can see in this example that the common factors of 8 and 12 were two times 2. The rest of the numbers 2 and 3 were multiplied with these common factors to obtain the LCM of 8 and 12 as 24.
Let us take another example of LCM of 510 and 92. Since these are two big numbers we can understand this even better with this example.
Here in this example of 510 and 92, we see that 2 is the only common prime factor. So, all the other factors are multiplied by 2 to get the LCM as 23460, which is a big number because there are not many common prime factors. So, by looking at the values of LCM we can find how closely the numbers are related with respect to prime factors.
Now let us calculate the LCM of three numbers by prime factorization. For example let us find out the LCM of 144, 180, and 192 by the prime factorization method.
Here you can see that we did not solve by taking the common factors into consideration. When there are more than 2 numbers we can solve by taking the maximum times a prime factor is repeated in a number and multiplying them. For example, in 144, 180, and 192, the maximum number of times 2 appears is in 192 (5 times), 3 appears a maximum of 2 times and 5 appears only one time. So we multiply them together to get the LCM as 2880.
LCM by long division
In this method, we divide both the numbers together with the lowest common factor, and we continue this process till we reach the stage where there no common factor to divide both the numbers. Let us again find the LCM of 8 and 12 by the long division method.
We can see that the common divisors can divide only two times after which we end up with prime factors 3 and 2. Now, we multiply the common factors 2,2 with reminders 3 and 2. This gives us an LCM of 24 for 8 and 12.
Now let us see another example with LCM of 336 and 54.
We can again after two divisions we end up with 56 and 9 and they do not have a common factor. So we multiply the common factors 2,3 with 56 and 9. So, the LCM of 336 and 54 is 3024.
LCM and HCF formula
The product of two numbers should be equal to the product of their LCM and HCF. This formula can be used for verification purposes to see if our answer is correct.
Let us take a simple example of 8 and 12.
- The LCM of 8 and 12 is 24
- The HCF of 8 and 12 is 4
- The product of LCM and HCF is 24 * 4 = 96
- The product of the two numbers is 8 * 12 = 96
- Hence proved the product of LCM and HCF = product of the two numbers
From the above formula we can deduce the formula for HCF and LCM as
HCF = product of numbers/LCM
LCM = product of numbers/HCF
Difference between LCM and HCF
| LCM | HCF |
| Least common multiple | Highest common factor |
| Value is greater than HCF | Value is lesser than LCM |
| Both the numbers can divide the LCM exactly | The HCF can divide both the numbers exactly |
| LCM = product of numbers/HCF | HCF = product of numbers/LCM |
HCF and LCM questions
Question: Find the HCF and LCM of 404 and 96.
The HCF of 96 and 404 is 2
The LCM of 96 and 404 is 6432
Question: Find the LCM and HCF of 90 and 144.
The HCF of 90 and 144 = 18
The LCM of 90 and 144 = 6480
Question: Find the LCM and HCF of 144 180 and 192 by prime factorization method
The HCF of 144, 180 and 192 = 12
The LCM of 144, 180 and 192 = 2880
Question: LCM of 12 and 18 by long division
The LCM of 12 and 18 is 36
Question: LCM of 510 and 92 by long division
The LCM of 510 and 92 is 23460
Question: LCM of 24 and 36 by long division
The LCM of 24 and 36 is 72
Question: LCM of 20 and 15 by long division
The LCM of 20 and 15 is 60
Let us find the HCF for verification of our answer:
- 20 = 2 * 2* 5
- 15 = 3 * 5
Hence, the HCF of 20 and 15 is 5
We can verify this answer by using the formula of HCF and LCM.
- Product of numbers = 20*15 = 300
- Product of LCM and HCF = 60 * 5 = 300
- Hence verified
Are online tools available for calculation for HCF and LCM?
Yes, online calculators are available for the calculation of LCM and HCF. But it is always better to calculate manually for small numbers. For large numbers, yes, you will have to use calculators. The following websites and apps can be useful:
HCF and LCM calculator android app
