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__HCF and LCM Important Formulas, Shortcut Methods and Tricks:__

__HCF and LCM Important Formulas, Shortcut Methods and Tricks:__

**Factors:**The numbers which exactly divide a given number are called the factors of that number. For example, factors of 15 are 1, 3, 5 and 15.

**Common Factor:**When a unique number divides exactly two or more given numbers then that number is called the common factor of the given numbers. For example, each of the numbers 2, 3 and 6 is a common factor of 12 and 18.

**Multiple:**When a number is exactly divisible by another number, then the former number is called the multiple of the latter number. Obviously, latter number is contained in the former. For example, 24 is a multiple of 1, 2, 3. 4, 6, 8, 12 and 24.

**Prime Number**: A number is said to be a prime number if its factors are 1 and the number itself. For example, 2, 3, 5, 7. 11. 13, 17, 19, 23, 29, 31 are prime numbers.

**Co-prime:**Two numbers are said to be co-prime (prime to each other) if there is no common factor other than 1 between them. It should be noted that the numbers which are co-prime, are not necessarily prime numbers. Even composite numbers may be mutually prime. For example, 5 and 7 are co-prime. Again, 16 and 25 are also co-prime where neither of them is a prime number.

**Least Common Multiple (L.C.M.):**The lowest common multiple (LCM) of two or more given numbers is the least number which is exactly divisible by each of them.

For example;

Multiples of 12 are 12, 24, 36, 48, 60, 72, ……..

Multiples of 18 are 18, 36, 54. 72, ........

Common multiples of 12 and 18 are 36, 72.

∴ Least Common Multiple (LCM) = 36

__How to Find Least Common Multiple (LCM)__:

**1. Factorization Method:**

Find the L.C.M. of 12, 27 and 40

Factors of 12 = 2 × 2 × 3 = 2² × 3

Factors of 27 = 3 × 3 × 3 = 3

^{3}
Factors of 40 = 2 × 2 × 2 × 5 = 2

^{3}× 5
∴ L.C.M. = 2

^{3}× 3^{3}× 5 = 1080**2. Short Cut Method (Division Method):**

Find the L.C.M. of 12, 27, and 40

∴ L.C.M = 2 × 2 × 3 × 9 × 10 = 1080

**Highest Common Factor (HCF):**The highest common factor of two or more numbers is the greatest number which divides each of them exactly. The HCF of two co prime numbers is 1. It can be taken as a test to decide whether two given numbers are co-prime or not.

For example,

Let’s consider two numbers 24 and 36.

All possible factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24

All possible factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

The common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12

The greatest common factor is 12.

Hence, 12 is the HCF of 24 and 36.

__How to Find Highest Common Factor (HCF)__:

**1. Factorization Method:**

Find H.C.F. of 48, 108, and 140

Factors of 48 = 2 × 2 × 2 × 2 × 3 = 2

^{4}× 3
Factors of 108 = 2 × 2 × 3 × 3 × 3 = 2² × 3

^{3}
Factors of 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7

∴ H.C.F. = 2² = 4

**Division Method:**

Find the H.C.F. of 48, 108, and 140

∴ H.C.F.=2 × 2= 4

__Some Important Results__:- If the numbers are divided by their HCF, then relatively prime numbers are obtained as quotient. Hence, if the numbers be
*‘x’*and ‘*y’*and their HCF be m, then*x = ma*and*y -*mb, where ‘a’ and ‘b’ are co-prime. - When two numbers divided by a third number leave the same remainder, then the difference of two numbers is exactly divisible by the third number. Accordingly the HCF of the numbers will be same as that of their differences.
- The HCF of the sum of the numbers and their LCM is equal to the HCF of the numbers.
- The product of two numbers is equal to the product of their HCF and LCM. i.e., First number × second number = HCF × LCM

L.C.M. of two numbers = Product of numbers / H.C.F. of numbers

L.C.M. of given fractions = L.C.M of numerators/H.C.F of denominators

H.C.F of given fractions = H.C.F of numerators/L.C.M of denominators

5. The L.C.M of a given set of numbers would be either the highest or higher than the highest of the given numbers.

6. The H.C.F. of a given set of numbers would be either the lowest or lower than the lowest.

__Quick & Shortcut Method:__

__Question:__Find the H.C.F. of 777 and 1147

__Solution:__

__Example: 01__**Find the HCF of 48, 168 and 324**

**Firstly, we find the HCF of 48 and 168.**

__Solution__:
∴ HCF of 48 and 168 =24

Now, we find
the HCF of 24 and 324.

∴ HCF of 48, 168 and 324 is 12.

__Example: 02__**Find the L.C.M. of (1/3), (5/6), (5/9) and (10/27)**

**L.C.M of fractions = L.C.M of numerators/H.C.F of denominators**

__Solution__:
L.C.M of 1,
5 and 10 is 10

L.C.M of 3,
6, 9 and 27 is 3

∴ L.C.M of given fraction is 10/3

__Example: 03__**Find the H.C.F. of (1/2), (3/4), (5/6), (7/8) and (9/10)**

**H.C.F of fractions = H.C.F of numerators/L.C.M of denominators**

__Solution__:
H.C.F. of 1,
3, 5, 7 and 9 is 1

L.C.M of 2,
4, 6, 8 and 10 is 120

∴ H.C.F. of given fractions = 1/120.

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