Question

Find the HCF of 105 and 1515 by prime factorisation method and hence, find its LCM.

Answer 1

Hint: In this question, we are given two numbers and we have to find their highest common factor using prime factorisation method. In the prime factorisation method, we will find prime factors of the given numbers and then identify common factors among them. Then, we will multiply the common factors obtained to find the highest common factor. To find the least common multiple, we will use prime factors obtained earlier and multiply all prime numbers occurring in prime factorisation of both numbers taking common prime factors only once.

Complete step by step answer:
Here, we are given two numbers which are 105 and 1515. Let us find their prime factorisation one by one.
For 105:
As we know, 3 is a prime and can be divided by 105 to get 35. So, 3 is one of the prime factors. For remaining numbers we can see 5 divides 35 to get 7. Since both 5 and 7 are prime numbers so they are also prime factors of 105.
Prime factorisation of $105=3\times 5\times 7$
For 1515:
As we know, 3 is a prime and can divide 1515 to get 505. So, 3 is one of the prime factors of 1515. For remaining number 505, we can see 5 divides 505 to get 101. Since 5 and 101 are both prime numbers, so they are also prime factors of 1515.
Prime factorisation of $1515=3\times 5\times 101$
Now,
\[\begin{align}
  & 105=3\times 5\times 7 \\
 & 1515=3\times 5\times 101 \\
\end{align}\]
We can see, 3 and 5 are the two common factors of 105 and 1515. To find the highest common factor we will multiply obtained common factors. Hence, common factor $3\times 5=15$. Therefore, HCF = 15
To find the least common multiple, we will multiply all the prime factors of both numbers taking common factors only once. As we can see, 3 and 5 are common factors in both numbers. So, we will take them only once. Other numbers that are left are 7, 101. So we will multiply all these 4 numbers to find the least common multiple.
Therefore, least common multiple $3\times 5\times 7\times 101=10605$. Hence, LCM = 10605
Therefore, HCF = 15 and LCM = 10605

Note: Students should take care that while finding least common multiple, they have to take common factors only once. Another method for finding LCM is by finding product of the numbers given and then divide by HCF obtained that is \[\text{LCM of two numbers}=\dfrac{\text{Product of two numbers}}{\text{HCF of two numbers}}\]
As we have obtained earlier HCF = 15, since, product can be obtained as $105\times 1515$. Therefore,
\[\text{LCM of two numbers}=\dfrac{105\times 1515}{\text{15}}=10605\]
So, students can use any of the two methods to find LCM.