Question

Find the LCM of $63$ and $105$ using prime factorisation method.

Answer 1

Hint: Here in this question, we have to find the HCF (Highest common factor) and LCM. The LCM is the least common multiple and it is defined as \[LCM(a,b) = \dfrac{{\left| {a \cdot b} \right|}}{{\gcd (a,b)}}\] , where a and b are integers and \[\gcd \]is the greatest common divisor. But here we use a division method. The HCF is the largest or greatest factor common among the two or more given numbers. To find the HCF and LCM we must have to find out its prime factors. Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number.

Complete step by step solution:
Now we find the LCM of 63 and 105, so we use the formula to find the LCM.
 Now we use the division method to find the LCM of given numbers so we have

3	63	105
7	21	35
3	3	5
5	1	5
	1	1

First, we divide the number by 3 if the number divides by 2 then we write the quotient otherwise we write the same number in the next line. Next, we will divide the numbers by 7 and the same procedure of writing is carried out. Again, next we divide by 3 and the same as above. Again next we will divide the numbers by 5 and hence we obtain 1 in the last row. This is the end od of the division procedure. We have to divide till we get 1 in the next row.
Now to find LCM of the given numbers we have to multiply the first column numbers that is
\[LCM = 3 \times 7 \times 3 \times 5\]
\[ \Rightarrow LCM = 315\]
Therefore, the LCM of 63 and 105 is 315.

Note: To solve these types of sums, one should be clear about the concept of HCF and LCM and the prime numbers. HCF is the highest or greatest common multiple whereas the LCM is the least common multiple or least common divisor in two or more given numbers. Prime numbers are the numbers greater than $1$ and which are not the product of any two smaller natural numbers. For Example: $2,{\text{ 3, 5, 7,}}......$ $2$ is the prime number as it can have only $1$ factor. Factors are the number $1$ and the number itself. Also, remember that we get the prime factorization of any composite number.