HCF and LCM notes for competitive exams

HCF and LCM

HCF & LCM are acronym for words, Highest common factor and Lowest common multiple respectively.

1. H. C. F

While we all know what a multiplication is like 2 * 3 = 6. HCF is just the reverse of multiplication which is known as Factorization. Now factorization is breaking a composite number into its prime factors. Like 6 = 2 * 3, where 6 is a composite number and 2 & 3 are prime number.
“In mathematics, the Highest Common Factor (HCF) of two or more integers is the largest positive integer that divides the numbers without a remainder. For example, the HCF of 8 and 12 is 4.”

Calculation

- By Prime Factorizations
Highest Common Factor can be calculated by first determining the prime factors of the two numbers and then comparing those factors, to take out the common factors.
As in the following example: HCF (18, 42), we find the prime factors of 18 = 2 * 3 * 3 and 42 = 7 * 2 * 3 and notice the "common" of the two expressions is 2 * 3; So HCF (18, 42) = 6.
- By Division Method
In this method first divide a higher number by smaller number.

• Put the higher number in place of dividend and smaller number in place of divisor.
• Divide and get the remainder then use this remainder as divisor and earlier divisor as dividend.
• Do this until you get a zero remainder. The last divisor is the HCF.
• If there are more than two numbers then we continue this process as we divide the third lowest number by the last divisor obtained in the above steps.

First find H.C.F. of 72 and 126
72|126|1
72       
54| 72|1              54              18| 54| 3
                    54                      0 
H.C.F. of 72 and 126 = 18

2. L.C.M

The Least Common Multiple of two or more integers is always divisible by all the integers it is derived from.  For example, 20 is a multiple of 5 because 5 × 4 = 20, so 20 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 4.
LCM cam also be understand by this example:
Multiples of 5 are:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70 ...
And the multiples of 6 are:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...
Common multiples of 5 and 6 are:
30, 60, 90, 120, ....
Hence, the lowest common multiple is simply the first number in the common multiple list i.e 30.

Calculation

- By Prime Factorizations
The prime factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers.
Example: To find the value of LCM (9, 48, and 21).
First, find the factor of each number and express it as a product of prime number powers.
Like 9 = 3²,
48 = 2⁴ * 3
21 = 3 * 7
Then, write all the factors with their highest power like 3², 2⁴, and 7. And multiply them to get their LCM.
Hence, LCM (9, 21, and 48) is 3² * 2⁴ * 7 = 1008.
- By Division Method
Here, divide all the integers by a common number until no two numbers are further divisible. Then multiply the common divisor and the remaining number to get the LCM.
2 | 72, 240, 196  2 | 36, 120, 98   2 | 18, 60, 49  3 | 9, 30, 49      | 3, 10, 49
L.C.M. of the given numbers
= product of divisors and the remaining numbers
= 2×2×2×3×3×10×49
= 72×10×49 = 35280.

Relation between L.C.M. and H.C.F. of two natural numbers

The product of L.C.M. and H.C.F. of two natural numbers = the product of the numbers.
For Example:
LCM (8, 28) = 56 & HCF (8, 28) = 4
Now, 8 * 28 = 224 and also, 56 * 4 = 224

HCF & LCM of fractions:

Formulae for finding the HCF & LCM of a fractional number.
HCF of fraction = HCF of numerator / LCM of denominator
LCM of Fraction = LCM of Numerator / HCF of Denominator