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Find the LCM and HCF of the following integers by applying the prime factorization method:
84, 90 and 120
Given: 84, 90 and 120.
To find: Here we have to find the LCM and HCF of the given integers by applying the prime factorization method.
Solution:
Calculating LCM and HCF using prime factorization method:
Writing the numbers as a product of their prime factors:
Prime factorisation of 84:
- $2\ \times\ 2\ \times\ 3\ \times\ 7\ =\ 2^2\ \times\ 3^1\ \times\ 7^1$
Prime factorisation of 90:
- $2\ \times\ 3\ \times\ 3\ \times\ 5\ =\ 2^1\ \times\ 3^2\ \times\ 5^1$
Prime factorisation of 120:
- $2\ \times\ 2\ \times\ 2\ \times\ 3\ \times\ 5\ =\ 2^3\ \times\ 3^1\ \times\ 5^1$
Multiplying the highest power of each prime number these values together:
$2^3\ \times\ 3^2\ \times\ 5^1\ \times\ 7^1\ =\ 2520$
LCM(84, 90, 120) $=$ 2520
Multiplying all common prime factors:
$2^1\ \times\ 3^1\ =\ 6$
HCF(84, 90, 120) $=$ 6
So, the LCM and HCF of 84, 90 and 120 are 2520 and 6 respectively.
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