**GETTING STARTED**

**HCF:**

☛ HCF stands for Highest Common Factor. As the name suggests, it is the largest number which divides all of the given numbers.

☛ Let us take two numbers, 12 and 18.

2, 3, 6 are the numbers which divide them both (common factors). Hence the HCF of the two numbers is 6.

**LCM:**

☛ LCM stands for Least Common Multiple. As the name suggests, it is the smallest number which is a multiple of all the given numbers.

☛ Let us take two numbers, 12 and 18.

36, 72, 108 are a few numbers which are multiples of both 12 and 18. But since 36 is the least of all, it is the LCM of given numbers.

**Method to find the HCF for given numbers:**

**Prime factorization Method:**

☛ Break down all the numbers into their corresponding prime factors raised to their powers

☛ Select the distinct common prime factor with least power and multiply all of them.

**Eg. 2160, 3420**

** 2160** = 8 x 27 x 10

= 2^{3 }x 3^{3 }x 2^{1 }x 5^{1 }

**= 2 ^{4 }x 3^{3 }x 5^{1}**

**3420** = 2 x 171 x 10 = 2 x 9 x 19 x 10

= 2^{1} x 3^{2} x 19^{1} x 2^{1} x 5^{1}

= **2 ^{2} x 3^{2} x 5^{1} x 19^{1}**

^{}

**Note:** In the above process of splitting numbers into their prime factors, we started off with numbers which are their factors for sure and then made them into smaller numbers further finding their other factors.

The **common factor is: 2 ^{2} x 3^{2} x 5^{1}**

Therefore the

**HCF = 4 x 9 x 5 = 180**

**Method to find the LCM for given numbers:**** Prime factorization Method:**

☛ Break down all the numbers into their corresponding prime factors raised to their powers

☛ Select the distinct common prime factors with maximum power (among the factors of given numbers) and multiply all of them.

**Eg. 2160, 3420**

2160 = 8 x 27 x 10

= 2^{3 }x 3^{3 }x 2^{1 }x 5^{1 }

= **2 ^{4 }x 3^{3 }x 5^{1}**

3420 = 2 x 171 x 10 = 2 x 9 x 19 x 10

= 2^{1} x 3^{2} x 19^{1} x 2^{1} x 5^{1}

= **2 ^{2} x 3^{2} x 5^{1} x 19^{1}**

**Least Common Multiple = 2 ^{4 }x 3^{3 }x 5^{1 }x 19^{1}**

**= 41040**

**LCM x HCF (of two numbers) = Product of the two numbers**

[This property can be applied only when we have 2 numbers]

**Properties of HCF & LCM**

☛ LCM of numbers is always divisible by HCF of numbers

☛ Difference between given numbers has to be divisible by HCF

☛ **If N is largest number which leaves the same remainder in dividing 3 different numbers – A, B and C,**

**then value of N = HCF [(middle number – smallest number), (largest number – middle number)]**

**Eg.** **Find the largest number which leaves the same remainder when dividing 49, 81 and 177 ^{}**

**Soln:** Difference between 49 and 81 = 81-49 = 32

Difference between 81 and 177 = 177 – 81 = 96

HCF = 32

**Therefore the highest number that will leave a common remainder is 32.**

The same property can be extended to any number of given numbers

☛ If N is largest number which leaves the same remainder in dividing M different numbers,

value of N = HCF [(second smallest number – smallest number), (third smallest number- second smallest number), …(largest number – second largest number)]

☛ LCM finds applications in problems where different actions are repeated at some different time intervals and we are asked to find at what time, all of them will coincide.