Page 136 - Ganit Kaushal

Step 3: Divide all the quotients obtained in step 2 by a common divisor and again get the quotients.

Step 4: Repeat this process until there are no common factors left, and multiply all the common divisors to
get the HCF of the given numbers.

Ex. Find the HCF of 12, 20 and 56.
Sol. Following the algorithm to find HCF of 12, 20 and 56.
2 12, 20, 56
We have, 2 6, 10, 28 ; we get 2 times 2 as the common factors.
3, 5, 14

\ HCF of 12, 20 and 56 = 2 × 2 = 4.
Common Multiples and Lowest Common Multiple

Again, recalling that multiples of a number can be obtained by multiplying it with natural numbers 1, 2, 3, 4,
5, and so on.

For example: Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, …
Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, …
We observe that the numbers 6, 12, 18, … etc., appear in the lists of multiples of both 2 as well as of 3. So,
they are called common multiples of 2 and 3.
Therefore, common multiples of two or more numbers are the numbers which appear in the lists of multiples
of all of them.
The Lowest Common Multiple (LCM) of two or more numbers is the lowest of all their common multiples.
For example: LCM (2, 3) = 6, since 6 is the lowest of all multiples 6, 12, 18, … of 2 and 3.

Algorithm to Find LCM by Common Division

Step 1: Write the given numbers in a row.
Step 2: Divide by the least prime number which divides at least one of the given numbers and carry forward
the numbers which are not divisible.
Step 3: Continue this process till we get only 1 in the last row.
Step 4: The product of the divisors is the required LCM of the given numbers.

Ex. Find the LCM of 24, 60 and 75.
Sol. Following the algorithm to find the LCM of 24, 60 and 75.
2 24, 60, 75
We have, 2 12, 30, 75
2 6, 15, 75
3 3, 15, 75
5 1, 5, 25
5 1, 1, 5
1, 1, 1 Here, the divisors are 2, 2, 2, 3, 5, 5.
\ LCM of 24, 60 and 75 = 2 × 2 × 2 × 3 × 5 × 5 = 600

Relation between HCF and LCM

There is a very important relation between the HCF and LCM of two given numbers, which is:

Product of any two numbers = Product of their HCF and LCM

134 Mathematics-6