HCF and LCM

In mathematics, while doing arithmetic calculations most of the time we need to find HCF and LCM which is very easy in case of small numbers. But when we take large numbers or group of numbers then it somewhat complicated. So to make it easy and simple we shall learn some easy tricks or methods to find the LCM and HCF which is like playing with numbers only.

HCF or GCD:

The abbreviation itself tells about the definition, HCF or GCD is the Highest Common Factor or Greatest Common Divisor of the given numbers which may be two or more numbers, and it is the largest number which divides the given numbers completely with remainder as zero.

For example:

  • HCF or GCD of 24 and 36 is 12, because 12 is the largest number which divides the numbers 24 and 36 without leaving remainder i.e. completely.
  • Also, the HCF or GCD of 4 and 6 is 2, because 2 is the largest number which divides the numbers 4 and 6 are completely.
  • In this we can find the HCF, we see some methods to find the HCF which are discussed as follows.

LCM:

The word LCM tells about it that is LCM is the least common multiple of given numbers. LCM is the smallest or least common multiple of given numbers.

For example:

The LCM of 2 and 3 is 6. That means 6 is the least common number which is divisible by both 2 and 3.

 

There are two methods by which we can easily find the HCF and LCM of given numbers.

These methods are

  1. Prime factorization method
  2. Division method

Find HCF by Prime factorization method :

In this method we have to find first the prime factors of the given numbers and then the common factors are taken of both the numbers. The product of the common factors gives the HCF of given numbers.

For example:

  • We have to find HCF of 24 and 36, then first we find the prime factors of the 24 and 36 as follows.

24 = 2*4*3 = 2*2*2*3

And 36 = 12*3= 4*3*3 = 2*2*3*3

Thus, the common prime factors of 24 and 36 are 2, 2 and 3.

Hence, HCF = 2*2*3= 12

Thus, HCF of 24 and 36 is 12.

 

  • If we have to fond the HCF of numbers 78, 84 and 96 at a time then we use some other trick for this also, let us see below.

First we find the prime factors of 78, 84 and 96 as follows.

78 = 2*39 = 2*3*13

84 = 2*42= 2*7*6= 2*7*3*2= 2*2*3*7

96 = 2*48 = 2*2*24 = 2*2*2*12 = 2*2*2*2*6 = 2*2*2*2*2*3

In this way we have find the prime factors, and we have taken the common factors of 78, 84 and 96 which are 2 and 3 only.

Now to find the HCF we take the product of these common factors.

Thus, HCF of 78, 84 and 96 = 2*3 = 6.

Find LCM by Prime factorization method :

In this prime factorization method, first we have to find all the prime factors of the given numbers.

Then, we have to find common prime factors and uncommon prime factors from both the numbers.

And finally, LCM = product of common prime factors and uncommon prime factors

i.e. LCM = (common prime factors)*(Uncommon prime factors)

For example:

  • If we have to fine the LCM of 16 and 30 then first we find their all prime factors.

16 = 2*8 = 2*2*4= 2*2*2*2

And 30 = 2*15 = 2*3*5

Thus, 16 and 30 are factorized completely into prime numbers.

Now we have to find the common prime factors which is 2 only here.

And the uncommon prime factors are 2,2,2,3 and 5

Thus, LCM of 16 and 30 = common prime factors*uncommon prime factors

LCM of 16 and 30 = 2*(2*2*2*3*5) = 2*(24*5) = 2*120 = 240

Thus, the LCM of 16 and 30 is 240.

 

  • Similarly, we can find the LCM of 36 and 60 as follows.

First we find the prime factors of 36 and 60 as follows.

36 = 2*18 = 2*2*9= 2*2*3*3

And 60 = 2*30 = 2*2*15 = 2*2*3*5

Thus, 36 and 60 are completely factorized into the prime numbers.

The common prime factors are 2, 2 and 3.

The remaining uncommon factors are 3 and 5.

Thus, LCM of 36 and 60 = common prime factors*uncommon prime factors

LCM of 36 and 60 = (2*2*3)*(3*5) = 12*15= 180.

Thus, the LCM of 36 and 60 is 180.

Find HCF by Division method :

In this division method, we have to divide by the lowest number to the larger one. Then again the remainder we have taken as divisor and by taking it as divisor we divide the first divisor we have taken.

In this way we proceed the same method till we get the remainder zero. Finally the divisor which makes remainder zero is taken as the HCF.

To understand clearly see the following example.

For example:

  • If we have to find the HCF of 24 and 36.

Now, first we take smaller number 24 as divisor. Divide 36 by 24 then we will get the remainder as 12.

Now, we take divisor which is the remainder 12, and we divide by 12 to the first divisor we have taken which is 24.

Thus, we divide 24 by 12 and our remainder will be zero.

Hence, the last divisor 12 which makes our remainder zero is our HCF.

Hence, HCF of 24 and 56 is 12.

  • Now, if we have to find the HCF of three numbers 78, 84 and 96.

Then first find the HCF of any two numbers as mentioned in above example.

Let, we find first the HCF of 78 and 84. Here, 78 is greater than 84, hence we divide 84 by 78.

Then the remainder will be 6. Now we take 6 as divisor and we divide by 6 to previous divisor which is 78. Thus, we divide 78 by the 6 and the remainder we will get is zero.

Hence, the HCF of 78 and 84 will be 6.

Now, we take this HCF 6 and the number remained which is 96.

Now we find the HCF of 6 and 96.

We divide 96 by 6 then the remainder we will get is zero directly.

Hence, 6 is the final HCF.

Thus, the HCF of 78, 84 and 96 is 6.

Find LCM by Division method :

In this division method, while finding the LCM we divide each number by least prime number simultaneously till the divisors becomes one.

And we take the product of that least prime numbers we have taken which is the LCM.

To understand more clearly see the following examples.

For example:

  • If we have to find the LCM of 16 and 30, then first we divide both by least prime number which is 2.

After dividing by 2 we get the divisors as 8 and 15.

Again we divide 8 and 15 by least prime number 2, then we get the divisors as 4 and 15.

Again we divide the 4 and 15 by least prime number 2, then we get the divisors as 2 and 15.

Again we divide the numbers 2 and 15 by least prime number 2, then the divisors are 1 and 15.

Now we divide to only 15 by least prime number 3, then the divisor is 5.

Again we divide 5 by least prime number 5 and finally both the divisors will be 1.

Hence, the LCM is the product of least prime numbers we have taken.

Hence, LCM of 16 and 30 = 2*2*2*2*3*5= 240

Hence, the LCM of 16 and 30 is 240.

Similarly, we can find the LCM of 36 and 60 as follows.

And the LCM we got is 180.


Updated: July 14, 2021 — 11:40 pm

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