In this section we will discuss about finding hcf lcm of positive integers.
In order to find the HCF and LCM of two or more positive integers, we may use the following steps. Step 1 : Factorize each of the given positive integers and express them as a product of powers of prime in ascending order of magnitudes of primes. Step 2 :To find the HCF, identify common prime factors and find the smallest exponent of these common factors. Now raise these common prime factors to their smallest exponents and multiply them to get HCF.
To find the LCM, list all prime factors occurring in the prime factorization of the given positive integers.
For each of these factors, find the greatest exponent and raise each prime factor to the greatest exponent and multiply them to get the LCM.
Examples on Finding Hcf Lcm of Positive Integers :
1) Find the HCF and LCM of 90 and 144 by the prime factorization method.
Solution :
90 = 2 x 45
= 2 x 3 x 15
= 2 x 3 x 3 x 5
90 = 2 x 3^{2} x 5
144 = 2 x 72
= 2 x 2 x 36
= 2 x 2 x 2 x 18
= 2 x 2 x 2 x 2 x 9
= 2 x 2 x 2 x 2 x 3 x 3
144 = 2^{4} x 3^{2}
To find the HCF, we list the common prime factors and their smallest exponents in 90 and 144 as under :
Common factors
Least exponents
2
1
3
2
∴ HCF = 2^{1} x 3^{2} = 2 x 9 = 18
To find the LCM, we will list all prime factors of 90 and 144 and their greatest exponents as follows :
Prime factors of 90 and 144
Greatest exponents
2
4
3
2
5
1
∴ LCM = 2^{4} x 3^{2} x 5 = 16 x 9 x 5 = 720
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2) Find the HCF of 96 and 404 by prime factorization method. Hence, find their LCM. Solution :
96 = 2^{5} x 3 and 404 = 2^{2} x 101
∴ HCF = 2^{2} = 4
⇒ LCM = (96 x 404) / HCF
⇒ LCM = (96 x 404)/4 = 96 x 101 = 9696. Euclid's Geometry