Let’s see LCM and HCF (The Least common multiple, The Highest Common Factor) definitions and methods for finding LCM and HCF step by step.

### Factors

If a number X divides another number Y exactly (without leaving any reminder), then X is a factor of Y and Y is a multiple of X.

Factors are the set of numbers that exactly divide the given numbers.

### Multiples

A set of numbers, which are exactly divisible by the given number, are multiplies of the given number.

For example:

If the given number is 8, then {8, 4, 2, 1} is the set of factors, while {8, 16, 24, 32, …} is the set of multiples of 8.

Note:

- Factors of a given number are always less than or equal to the given number.
- Multiples of a number are always more than or equal to the given number.
- 1 is the factor of every number.
- Every number is a factor and multiple of itself.

### Common Multiples

A common multiple of two or more numbers is a number that is completely divisible (without leaving a remainder) by each of them.

For Example:

We can obtain common multiples of 3, 5, and 10 as follows:

Multiples of 3 = {3, 6, 9, 12, 15, 18, 21, 24, 27, **30**, 33, …}

Multiples of 5 = {5, 10, 15, 20, 25, **30**, 35,…}

Multiples of 10 = {10, 20, **30**, 40, …}

Here we can see that in above factors common multiple of 3, 5, and 10 is 30.

Some another examples of common factors of 3, 5, and 10 = {30, 60, 90, 120, …}

## Least Common Multiple (LCM)

The LCM of two or more given numbers is the least number which is exactly divisible by each of them.

For Examples:

If we have to find out the LCM of 4 and 12, then we can find like this:

Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, …

Multiples of 12 = 12, 24, 36, …

Here we can see the common multiples of 4 and 12 are 12, 24, … and the least common multiple is 12.

Hence, the LCM of 4 and 12 is = 12.

## Methods to Calculate LCM

There are two methods to find the LCM of two or more numbers, which are explained below:

### 1. Prime Factorization Method.

Following are the steps to obtain LCM through the prime factorization method.

**Step I** Resolve the given numbers into the product of their prime factors.

**Step II** Find the product of all prime factors (with the highest powers) that occurs in the given numbers.

**Step III** This product of all the prime factors (with the highest powers) is the required LCM.

### 2. Division Method

Following are the steps to obtain LCM through Division Method.

**Step I** Write down the given numbers in a row, separating them by commas.

**Step II** Divide them by a prime number, which exactly divides at least two of the given numbers.

**Step III** Write down the quotient and the undivided numbers in a line below the first.

**Step IV **Repeat the process until you get a line of numbers that are prime to one another.

**Step V** The product of all divisors and the numbers in the last line will be the required LCM.

### Common Factor

A common factor of two or more numbers is that particular number that divides each of them exactly.

For Examples:

We can obtain common factors of 12, 22, and 30 as follows:

Factors of 12: 1, 2,3, 4, 6, 12

Factors of 22: 1, 2, 11, 22

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Here we can see the common factor of 12, 22, and 30 = 2.

## Highest Common Factors (HCF)

HCF of two or more given numbers is the greatest number which divides each of them exactly.

For Example:

6 is the HCF of 12 and 18. As there is no other greater number that can divide 12 and 18 exactly.

HCF is also known as Highest Common Divisor (HDC) and Greatest Common Measure (GCM).

## Methods To Calculate HCF

There are two methods to calculate the HCF of two or more numbers which are explained below:

### 1. Prime Factorization Method

Following are the steps for calculating the HCF of given numbers:

Step I Resolve the given numbers into products of their prime factors.

Step II Find the product of all the prime factors (with least power) common to all the numbers.

Step III The product of common prime factors (with the least power) gives HCF.

### 2. Division Method

Following are the steps to obtain HCF through the division method for two numbers:

Step I To find the HCF of two given numbers, divide the largest number by the smaller one.

Step II Divide the divisor of step I by the remainder obtained in step I.

Step III Repeat step II till the remainder becomes zero. The last divisor is the required HCF.

Note:*

To calculate the HCF of more than two numbers, calculate the HCF of the first two numbers, then take the third number and HCF of the first two numbers and calculate their HCF. The resulting HCF will be the required HCF of numbers.

Note*

To find the HCF of given numbers, we can divide the numbers by their lowest possible difference. If these numbers are divisible by this difference then this difference itself is the HCF of the given numbers, otherwise, any other factor of this difference will be its HCF.

## Methods To Calculate LCM And HCF of Fractions

The LCM and HCF of fractions can be obtained by the following formulae:

LCM of Fractions=\frac{LCM of Numerators}{HCF of Denomenators}HCF of Fractions=\frac{HCF of Numerators}{LCM of Denomenators}

**Note***

- All the fractions must be in their lowest terms. If they are not in their lowest terms, then conversion in the lowest form is required before finding the HCF or LCM.
- The required HCF of two or more fractions is the highest fractions, which exactly divides each of the fractions.
- The required LCM of two or more fractions is the least fractions/integer, which is exactly divisible by each of them.
- The HCF of numbers of fractions is always a fraction, but this is not true in case of LCM.

Solved Examples:

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