Formulas

THE CONCEPT OF GCD (GREATEST COMMON DIVISOR OR HIGHEST COMMON FACTOR)

• Consider two natural numbers n, and n2. If the numbers n1 and n2 are exactly divisible by the same number x, then x is a common divisor of n1 and n2.
• The highest of all the common divisors of n1 and n2 is called as the GCD or the HCF. This is denoted as GCD (n1, n2).

Rules for Finding the GCD of Two Numbers n1 and n2

(a) Find the standard form of the numbers n1 and n2.
(b) Write out all prime factors that are common to the standard forms of the numbers n1 and n2.
(c) Raise each of the common prime factors listed above to the lesser of the powers in which it appears in the standard forms of the numbers n1 and n2.
(d) The product of the results of the previous step will be the GCD of n1 and n2.

THE CONCEPT OF LCM (LEAST COMMON MULTIPLE)

Let n1, and n2 be two natural numbers distinct from each other. The smallest natural number n that is exactly divisible by n1 and n2 is called the Least Common Multiple (LCM) of n1 and n2 and is designated as LCM (n1, n2).

Rule for Finding the LCM of two Numbers n1 and n2

(a) Find the standard form of the numbers n1 and n2.
(b) Write out all the prime factors, which are contained in the standard forms of either of the numbers.
(c) Raise each of the prime factors listed above to the highest of the powers in which it appears in the standard forms of the numbers n1 and n2.
(d) The product of results of the previous step will be the LCM of n1 and n2.

Rule for Finding out HCF and LCM of Fractions

(A) HCF of two or more fractions is given by: $\dfrac{HCF of Numerator}{LCM of Denominators}$
(B) LCM of two or more fractions is given by: $\dfrac{LCM of Numerator}{HCF of Denominators}$

Rules for HCF:

If the HCF of x and y is G, then the HCF of (i) x, (x + y) is also G
(ii) x, (x – y) is also G

Least Common Multiple( LCM):

A common multiple is one that is a multiple of 2 or more than 2 numbers.

For example:

The common multiples of 2 and 3 are 6,12,18, etc.

The Least Common Multiple of 2 numbers is the smallest positive number that is a multiple of both.

In other words, the Least Common Multiple of 2 or more numbers is the smallest number which is divisible by all the given numbers.

Multiples of 2: 2,4,6,8…
Multiples of 3: 3,6,9,12…
LCM of 2 and 3 will be 6.

How To Find Out LCM:

Method 1: Prime Factorization Method

Factorize all numbers into their prime factors Make a note of all the distinct factors Raise each factor to the maximum power present and multiply them all

Example:

To find the LCM of 136,144,168
136 = 23 x17
144 = 24 x 32
168= 23x3x7 Distinct factors are 2, 17, 3 and 7.
The highest power of 2 is 4, of 3 is 2, of 17 is 1 and of 7 is 1. So, LCM = 24 x 32 x17 x7 = 17136

Method 2:

To calculate the LCM of 4, 5 and 6, take the highest number: 6 in this case.
Now start with the multiples of 6 and check whether they are the multiples of 4 and 5 or not.
The first common multiple (i.e. the multiple of all 3?4,5, and 6) will be the LCM.
You start with 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 So, 60 is the LCM as it is the first number to be divisible by 4,5 and 6.

Illustrations:

Exercise:

Highest Common Factor (HCF)

Greatest Common divisor (GCD), also called HCF, is the largest integer that perfectly divides two or more given numbers.

For Example:

The HCF of 18 and 12 is 6.

6 is the largest number that divides both 18 and 12.

How To Find Out HCF:

Method 1: Prime Factorization Method-

Factorize all numbers into their prime factors Make a note of all the distinct factors present in all three numbers Raise each factor to the minimum power present and multiply them all

Example:

To find the HCF of 136,144, 168
To find the HCF of 136,144, 168
Step 1: 136 = 23 x17 144 = 24 x 32 168= 23x3x7
Step 2: Distinct factors present = 2,3,7,17
Step 3: Raising each factor to the minimum present (i.e. 23,30,70 and 170) HCF= 23=8

Method 2: Division Method

To find the HCF of 2 numbers by Division Method, the higher number is divided by the lower number.
Then the lower number is divided by the remainder obtained in the previous division.
This remainder is divided by the next remainder and so on till the remainder is zero. The last divisor will be the HCF of the two numbers

Example:

To find the HCF of 12 and 15
To find the HCF of 12 and 15
15/12|R = 3 12/3|R=0
Thus, the HCF= 3

Formulas:

HCF and LCM of Fractions:

HCF of fractions = HCF of numerators/ LCM of denominators
LCM of fractions = LCM of numerators/ HCF of denominators
LCM x HCF = Product of two numbers (this can be applied only for 2 numbers)

Example 1:

HCF of 12 and 24 = 12. LCM of 12 and 24= 24 HCF x LCM = 12×24 = product of the two numbers
This formula can be applied to any number of numbers only if all the numbers are relatively prime.

Example 2:

HCF of 4,5 and 6 = 1. LCM of 4,5 and 6 = 120 HCF x LCM= 120×1 = 120 = Product of the numbers.

Exercise:

Properties of HCF and LCM:

The HCF of two or more numbers is lesser than or equal to the smallest of those numbers
The LCM of two or more numbers is greater than or equal to the greatest of those numbers
If a number X always leaves a remainder R when divided by the numbers A,B,C.., then X= LCM(or a multiple of LCM) of A,B,C…+R

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