HCF means Highest Common Factor. Now, What is the factor?

A factor is a number that divides the given number completely eg: 2 divides 4 completely therefore 2 is a factor of 4.

To find HCF of given numbers.

**Firstly**, we find factors of given numbers.

**In the 2nd step**, we write common factors of given numbers.

**And in the final step**, we write the Highest Common factor (HCF)

**Note**: The greatest number that divides the numbers completely is called HCF of the numbers.

**Methods of HCF**

**HCF by Factorisation method**

Let us take a example:-

Example 1: Find the HCF of 18 and 36

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

Factors of 36 = 1,2,3,4,6,9,12,18,36

Common Factors of 18 and 36 = 1,2,3,6,9,18.

Highest Common factor (HCF) = 18.

Since 18 is the highest number that divides both the numbers 18 and 36 completely.

**HCF by Prime Factorisation Method**

We take the same example as above:

Prime factors of 18 = 2 x 3 x 3

Prime factors of 36 = 2 x 2 x 3 x 3

Common factors = 2 x 3 x 3

Now, Product of common factors = HCF

∴ HCF = 18

**HCF by long division method**

Let us take an example:

Find HCF of 144 & 180 by long division method.

**Steps:**

- Divide the larger number by the smaller number.
- After division, we get quotient and remainder
- Now in the next step, the remainder becomes the new divisor & the original divisor becomes the new dividend.
- Repeat the above step again and continue the process till you get zero as a remainder.
- The last divisor in the process is the HCF of the numbers.

Let us take another example:

**Find HCF of 120 & 239**

So, HCF of 120 and 239 = 1

**Remember:**

1. HCF of two consecutive numbers is always 1.

e.g: HCF of 4 & 5 = 1, 9& 10 = 1 etc.

2. HCF of two consecutive even numbers is always 2.

e.g: HCF of 2 & 4 = 2 and HCF of 20 & 22 = 2 etc.

3. HCF of two consecutive odd numbers is always 1.

e.g: HCF of 11 and 13 = 1, HCF of 25 and 27 = 1 etc.

**LCM (Lowest Common Multiple)**

Lowest Common Multiple (LCM) of the two or more numbers is the smallest number that is completely divisible by the numbers.

Example: Find LCM of 3 and 6?

Multiples of 3 = 3,6,9,12,15,18,21,24,27,30…

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

Common multiples = 6,12,18,24,30…

Now 6 is the lowest common multiple

Lowest common multiple = 6.

**LCM by prime factorisation method**

Example: Find LCM of 6,8 and 12.

Prime factors of 6 = 2 x 3

Prime factors of 8 = 2 x 2 x 2

Prime factors of 12 = 2 x 2 x 3

= 2 x 2 x 3 x 2

= 12 x 2 = 24

= LCM of 6, 8 & 12 = 24

Note: Multiply the common factors & the factors left out.

**LCM by Long division method**

Example:

Find LCM of 60, 70, 108.

2 | 60 | 70 | 108 |

2 | 30 | 35 | 54 |

3 | 15 | 35 | 27 |

3 | 5 | 35 | 9 |

3 | 5 | 35 | 3 |

5 | 5 | 35 | 1 |

7 | 1 | 7 | 1 |

1 | 1 | 1 |

= 2 x 2 x 3 x 3 x 3 x 5 x 7

= 12 x 9 x 5 x 7

= 108 x 5 x 7

= 540 x 7

= 3780

So, LCM of 60, 70, 108 = 3780

**Relationship between HCF and LCM**

Numbers |
HCF |
LCM |
Product of Numbers |
Product of HCF & LCM |

6, 8 | 2 | 24 | 6 x 8 = 48 | 2 x 24 = 48 |

15, 30 | 15 | 30 | 15 x 30 = 450 | 15 x 30 = 450 |

40, 50 | 10 | 200 | 40 x 50 = 2000 | 10 x 200 = 2000 |

From the above table we conclude that the product of the numbers always equal to the product of their LCM & HCF.

Product of the number = LCM x HCF.

LCM = Product of two numbers ÷ HCF

HCF = Product of two numbers ÷ LCM