Question

Find the H.C.F and L.C.M of 861, 1353.

Answer 1

Hint: To find HCF and LCM of larger numbers, find its prime factorization to find all the prime factors of both the numbers. Now we will obtain the common prime factors, larger prime factors and smallest common prime factors of both the numbers. HCF of those numbers will be the product of common prime factors of both the numbers. LCM will be the product of all the distinct prime factors with its corresponding greatest number of occurrences among all prime factorizations.

Complete step-by-step answer:
First let us find the HCF of 861 and 1353. For that we need to find the prime factorization of both the numbers.
The prime factorization of 861 is $3 \times 7 \times 41$.
And the prime factorization of 1353 is $3 \times 11 \times 41$.
HCF of the two numbers is the product of their common prime factors. And the common factors are 3 and 41 which are present in both prime factorizations. Implies
HCF (861, 1353) = $3 \times 41 = 123$
Now we need to find the LCM of 861 and 1353 which is the product of all distinct prime factors of both eliminating the repetitions. Here the distinct factors in the two factorizations are 3, 7, 11, 41.
Thus LCM (861, 1353)=$3 \times 7 \times 11 \times 41 = 9471$

Therefore, The HCF of $861$ and $1353$ is $123$ and the LCM of $861$ and $1353$ is $9471$.

Note: Prime factorization of a number has to be done by checking all the prime factors are either a factor of given number or not. We start by the smallest prime number 2, then we check for 3 and then 5 etc. Thus for 861 checking from the smallest prime 2, which is not a factor of 861. As the next prime 3 and then 7 and then the remaining factor we obtain is 41. Thus 3, 7 and 41 are the three prime factors of 861. Similarly for 1353, we get 3 and 41 as prime factors.