Question

Find the LCM using the prime factorization method: \[48,64,72,96,108\].

Answer 1

Hint: In order to find the LCM of the numbers using prime factorization, first write the prime factors of the given numbers, find the least common multiple which is suitable for both the numbers. The number obtained must be the multiple of all the numbers \[48,64,72,96,108\] then only it will be called as their LCM.

Complete step by step solution:
We are given \[48,64,72,96,108\] to find their LCM, but first we should know what is LCM.
LCM stands for Least common multiple. A number which is the multiple of all the numbers listed to find the LCM.
The prime factors of the given numbers are:
\[48 = 2 \times 2 \times 2 \times 2 \times 3\]
\[64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
\[72 = 2 \times 2 \times 2 \times 3 \times 3\]
\[96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3\]
\[108 = 2 \times 2 \times 3 \times 3 \times 3\]
We can see that the six \[2's\] is maxm in the number’s, and others have two, three and four \[2's\]. So, for the least common multiple of all the numbers we can take the six \[2's\], which will automatically include all other’s requirements of \[2's\] in total.
Similarly, for \[3\], we have maxm three \[3's\], so for all the numbers in together, we are taking three \[3's\] in total and we get:
LCM of \[48,64,72,96,108 = 6\left( {2's} \right) \times 3\left( {3's} \right) = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 1728\]
Therefore, the Least Common Multiple (LCM) of \[48,64,72,96,108\] is \[1728\].

Note:
1) Do not include the least count or common number of \[2's\] and \[3's\] for the LCM, otherwise it would not give the correct answer.
2) We can also solve for LCM by simply listing some of the multiples of \[48,64,72,96,108\] then the first common multiple of both the numbers becomes the LCM.
For example, the multiple of two numbers are:
\[25 = 25,50,75,100,125,150,175,200,225.....\]
\[35 = 35,70,105,140,175,210,245....\]
3) We can see that the first common value obtained is \[175\], and this is our LCM for both the numbers.