Before starting the HCF and LCM topic it is important to understand what is a factor and what is a multiple this is because HCF or Highest Common Factor is related to the term factor and LCM or Least Common Multiple is related to the term Multiple.

Factors are a set of numbers which exactly divides any given number. For instance factors of 10 are 1, 2, 5 and 10 as they all divide the number 10 without leaving any remainder. Likewise, the factors of 20 are 1, 2, 4, 5, 10 and 20.

Multiples are a set of numbers which are exactly divisible any given number. For example, multiples of 5 are 5, 10, 15, 20 …

Alright, we now know what factors and multiples are so, it’s time for us to discuss about common factors and common multiples.

A common factor of two or more numbers is a number that exactly divides each of the given numbers.

For instance, find the common factors of 12, 16 and 18.

Factors of 12 = 1, 2, 3, 4, 6 and 12

Factors of 16 = 1, 2, 4, 8, and 16

Factors of 18 = 1, 2, 3, 6, 9, and 18

Therefore, common factors of 12, 16 and 18 are 1 and 2.

A common multiple of any two or more given numbers is a number which is completely divisible by each of the given numbers.

Find the common multiples of 4, 6 and 8.

Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 …

Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54 …

Multiples of 8 = 8, 16, 24, 32, 40, 48, 56 …

So, common multiples of 4, 6 and 8 are 24, 48 …

HCF of two or more numbers is the greatest or highest factor that is common to each of the given numbers.

For example, HCF of 12, 16 and 18 is 2.

LCM of two or more numbers is the least or smallest multiple that is common to each of the given numbers.

For example, LCM of 4, 6 and 8 is 24.

We general use the prime factorization method and the division method to find the HCF of given numbers. Let us discuss the prime factorization method first.

In this method we first express the given numbers as a product of all the prime factors. Then we multiply the common prime factors with least power to get the HCF. Here is an example that will clear the concept.

Find the HCF of 10, 20 and 30.

First we have to express the given numbers as the product of the prime factors.

10 in terms of its prime factors = 2x5

20 in terms of its prime factors = 2x2x5 = 2^{2}x5

30 in terms of its prime factors = 2x3x5

Now we will find the product of the common prime factors with least power. in this case the common prime factors with least power is 2x5 which is equal to 10.

Therefore, HCF of 10, 20 and 30 is 10.

In this method divide the largest number with the second largest number. Then we divide the divisor with the remainder. We repeat this process till the remainder becomes zero. The last divisor is the required HCF of the given numbers. Consider the following example.

Find the HCF of 10, 20 and 30 using the division method.

First we will divide the largest number 30 with the second largest number 20.

20 ) 30 ( 1

20

10 ) 20 ( 2

20

0

So, 10 is last divisor. Now we will divide 10 with the last divisor we just got. in this case 10 exactly divides 10 so we will get zero remainder.

Hence the HCF of 10, 20 and 30 is 10.

Like HCF we find LCM using two methods namely the prime factorization method and the division method. First we will look at the prime factorization method then we will discuss the division method.

In this method we first express the given numbers as a product of all the prime factors. Then we multiply all the prime factors with highest power to get the LCM. Consider the following example.

Find the LCM of 4, 6 and 8 using prime factorization method.

First we will express the given numbers in terms of prime factors.

4 in terms of its prime factors = 2x2 = 2^{2}

6 in terms of its prime factors = 2x3

8 in terms of its prime factors = 2x2x2 = 2^{3}

Now we will multiply all the prime factors with highest power. in this case the prime factors are 2 and 3 and the highest power are 3 and 1 respectively.

So, the require LCM of 4, 6 and 8 is 2^{3}x3 i.e. 24.

In the division method we take all the numbers and divide them together by the prime factors. Finally we multiply all the prime factors to get the required LCM of the given numbers.

Note! we start the division with the least prime factors.

Here is an example to find the LCM using division method.

Find the LCM of 4, 6 and 8 using division method.

2 |4, 6, 8

2 |2, 3, 4

2 |1, 3, 2

3 |1, 3, 1

|1, 1, 1

Product of all the prime factors = 2x2x2x3 = 24.

Therefore, the required LCM is 24.

- HCF of fractions = HCF of numerators

LCM of denominators - LCM of fractions = LCM of numerators

HCF of denominators - If A and B are two numbers then product of A and B is equal to the product of the HCF and LCM of the two numbers.

AxB = HCF(A,B) x LCM(A,B) - HCF of two prime numbers is always 1.

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