All about HCF AND LCM

All about HCF AND LCM:

Factor or Sub-multiple:
A number is said to be factor or sub-multiples of another number when it is exactly divides the other number i.e. the remainder obtained is zero.

Common Factor:
When unique number divides exactly two or more given numbers then that number is called the common factor of the given numbers. For example 5 is common factor of 15 and 45.

Multiple:
When a number is completely divisible by another number, then the former number is called the multiple of the later. Obviously, the former number contains the latter number. Example, 24 is a multiple of 1, 2, 3, 4, 6, 8, 12 and 24.

Co-Prime or Relatively Prime Numbers:
Two numbers are said to be co-prime i.e. prime to each other if there is no common factor except 1 in between them. Example: 5 and 7 are co-prime, 16 and 25 are co-prime.

HCF(Highest Common Factor):
Steps for finding HCF of the given numbers.

Step 1:
Find the standard form of the numbers.
150 = 5 × 5 × 3 × 2
210 = 5 × 2 × 7 × 3
375 = 5 × 5 × 5 × 3

Step 2:
Write all common factors that are common to the standard forms of the numbers.
5 × 3.

Step 3:
Raise each common factors listed above to the lesser of the power in which it appears in the standard forms of the numbers.
5 × 3.

Step 4:
The product of the result of the previous step will be HCF of the given numbers.
HCF = 15.

LCM(Least Common Multiple):
Steps for finding LCM of the given numbers.

Step 1:
Find the standard form of the number.
150 = 5 × 5 × 3 × 2
210 = 5 × 2 × 7 × 3
375 = 5 × 5 × 5 × 3

Step 2:
Write all Prime factors, which are contained in the standard forms of either of the numbers.
5, 2, 3, 7.

Step 3:
Raise each common factors listed above to the highest of the power in which it appears in the standard forms of the numbers.
= 2¹ × 3¹ × 5³ × 7¹.

Step 4:
The product of the result of the previous step will be LCM of the given numbers.
LCM = 2¹ × 3¹ × 5³ × 7¹ = 5250.

HCF and LCM of Fractions:
i. HCF of two or more fractions is given by : $$\frac{{{\text{HCF}}\,{\text{of}}\,{\text{Numerators}}}}{{{\text{LCM}}\,{\text{of}}\,{\text{Denominators}}}}$$

ii. LCM of two or more fractions is given by : $$\frac{{{\text{LCM}}\,{\text{of}}\,{\text{Numerators}}}}{{{\text{HCF}}\,{\text{of}}\,{\text{Denominators}}}}$$

KEY FACTS:
HCF of the numbers × LCM of the numbers = Multiplication of the numbers.

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