Question

What is the least common multiple of 6,9 and 10 ?

Answer 1

Hint: In this question , we need to find the least common multiple of the given three numbers. In order to find the least common multiple of the given numbers, we will use the method of prime factorisation. First we need to find the prime factors of the given three numbers. Then we can find the LCM of the given numbers by multiplying the prime factors of the numbers that is the maximum number of times of the occurrence of the prime factors in the given number.

Complete step by step answer:
Given numbers, \[6,\ 9\ \] and \[10\]
The least common multiple of \[6,\ 9\] and \[10\] is the smallest number
Among all common multiples of \[6,\ 9\], and \[10\].
Now we can list the prime factors of the given number,
Factors of \[6\] ,
\[\Rightarrow \ 2 \times 3\]
Factors of \[9\] ,
\[\Rightarrow \ 3 \times 3\]
Factors of \[10\] ,
\[\Rightarrow \ 2 \times 5\]
Now we need to multiply each prime factor , the greatest number of times it appears in any one factorization.
For the prime factor \[2\],
We have,
The number of times of occurrence of \[2\] in prime factorisation of \[6\ = \ 1\]
The number of times of occurrence of \[2\] in prime factorisation of \[9\ = \ 0\]
The number of times of occurrence of \[2\] in prime factorisation of \[10\ = \ 1\]
So, the maximum number of times of occurrence of \[2\ = \ 1\] ••• (1)
Similarly, for the prime factor \[3\],
We have,
The number of times of occurrence of \[3\] in prime factorisation of \[6\ = \ 1\]
The number of times of occurrence of \[3\] in prime factorisation of \[9\ = \ 2\]
The number of times of occurrence of \[3\] in prime factorisation of \[10\ = \ 0\]
So, the maximum number of times of occurrence of \[3\ = \ 2\] ••• (2)
Finally , for the prime factor \[5\],
We have,
The number of times of occurrence of \[5\] in prime factorisation of \[6\ = \ 0\]
The number of times of occurrence of \[5\] in prime factorisation of \[9\ = \ 0\]
The number of times of occurrence of \[5\] in prime factorisation of \[10\ = \ 1\]
So, the maximum number of times of occurrence of \[5\ = \ 1\] ••• (3)
From (1) , (2) and (3) ,we can say that \[2\] must occur once , \[3\] must occur twice and finally \[5\] must occur once .
Thus,
\[LCM = 2 \times 3 \times 3 \times 5\]
On multiplying,
We get,
\[LCM = 90\]
Therefore the LCM of \[6,9\] and \[10\] is \[90\] .

Note:
Here we need to notice that the terms \[9\] and \[10\] don’t have any common factors other than \[1\] . So the LCM of 9 and \[10\] is \[9\times10\] is \[90\] . We also see that \[90\] is divisible by \[6\] . So the LCM of \[6 , 9\] and \[10\] is \[90\] . The other methods used to find the least common factor are
1.By listing the multiples
2.By division method
LCM is known as the least common multiple. LCM of one or more given numbers is the least of their common factors.