HCF and LCM

Highest Common Factor (HCF)

Factor: Numbers which are multiplied together to get another number is called factors of that number. e.g. 3 and 4 are factors of 12, because 3 × 4 = 12. A number can have many factors.

Common factors: Factors that are common to two or more numbers are called common factors.

Highest Common Factor: HCF of two or more numbers is the greatest number that divides each of them exactly

How to find HCF of Given numbers:

Method I:

Factorisation or prime numbers method of finding HCF:

Step 1: Express each number as a product of prime factors.

Step 2: HCF is the product of all common prime factors using the least power of each common prime factor.

e.g.

Find the HCF of 27 and 36

27 = 3³

36 = 3²x 2²

Find the HCF of 108, 288 and 360.

108 = 2² x 3³

288 = 2⁵ x 3²

360 = 2³ x 5 x 3²

HCF = 2² x 3²= 36

Method II:

Division Method to find HCF (Shortcut):

Step 1: Divide the larger number by the smaller number

Step 2: Divisor of step 1 is divided by its remainder

Step 3: Divisor of step 2 is divided by its remainder. Continue this process till we get 0 as the remainder.

Step 4: Divisor of the last step is the HCF.

e.g. Find the HCF of 108 and 288.

How to find HCF of three numbers using division method

Step 1: Find out HCF of any two numbers.

Step 2: Find out the HCF of the third number and the HCF obtained in step 1

Step 3: HCF obtained in step 2 will be the HCF of the three numbers

Note: The Number that divides each of the two numbers also divide their sum, their difference and the sum and difference of any multiples of that numbers.

e.g HCF of 42 and 70

= HCF of 42 and 28(70-42)

= HCF of 28 and 14 (42-28)

= HCF of 14 and 14 (28-14)

= 14,

Hence, HCF = 14

(keep this method in mind, it gives result very quickly)

HCF of decimals:

Make the decimal place same in all the given numbers, find the HCF as they are integers, and adjust decimal accordingly in the final result.

e.g Find the HCF of 16.5, 0.45 and 15.

Solution:

numbers can be written as, 16.50, 0.45 and 15.00

now find the HCF of 1650, 45 and 1500, which is 15.
Adjust the decimal accordingly, i.e. 0.15

HCF of Fractions:

The HCF of two or more fractions is the highest fraction which is exactly divisible by each of the fractions.

HCF = (HCF of Numerators) / (LCM of denominators)

Least Common Multiple (LCM)

Multiple: Multiple is the product of any quantity and an integer.

Common Multiple: Common multiple is that which is multiple of two or more numbers. The common multiples of 2 and 3 are 0, 6,12,18, 24, etc.

Least Common Multiple (LCM) of two or more numbers is the smallest number (other than zero) that is a multiple of all the numbers

Example: LCM of 5 and 6 = 30 because 30 is the smallest multiple which is common to 5 and 6 (Or we can say that 30 is the smallest number which is divisible by both 5 and 6)

How to find LCM of Given numbers:

Method I:

Factorisation or prime numbers method to Find LCM:

Step 1: Express each number as a product of prime factors.
Step 2: LCM is the product of highest powers of all prime factors

Example 1: Find out LCM of 8 and 14

Step 1: Express each number as a product of prime factors.
8 = 2³

14 = 2 × 7

Step 2: LCM is the product of highest powers of all prime factors
Here the prime factors are 2 and 7
The highest power of 2 here = 2³
The highest power of 7 here = 7
Hence LCM = 2³ × 7 = 56

Method II:

Division Method to Find LCM (Shortcut):

Step 1: Write the given numbers in a horizontal line separated by commas.

Step 2: Divide the given numbers by the smallest prime number which can exactly divide at least two of the given numbers.

Step 3: Write the quotients and undivided numbers in a line below the first.

Step 4: Repeat the process until we reach a stage where no prime factor is common to any two numbers in the row.

Step 5: LCM is the product of all the divisors and the numbers in the last line.
e.g. Find the LCM of 20, 25 and 30 by long division method.