Find the LCM of the following using prime factorization method: 72, 90, 120.

To find the least common multiple (LCM) of the given numbers using prime factorization, we need to find the prime factorization of each number and then multiply the highest power of each prime factor.

Let's start by finding the prime factorization of each number:

Prime factorization of 72:
72
=
2
3
×
3
2
72=2
3
×3
2


Prime factorization of 90:
90
=
2
×
3
2
×
5
90=2×3
2
×5

Prime factorization of 120:
120
=
2
3
×
3
×
5
120=2
3
×3×5

Now, we can identify the highest power of each prime factor that appears among the three numbers:

The highest power of 2 is 3.
The highest power of 3 is 2.
The highest power of 5 is 1.
To find the LCM, we multiply the highest powers of the prime factors:

�
�
�
=
2
3
×
3
2
×
5
1
=
8
×
9
×
5
=
360
LCM=2
3
×3
2
×5
1
=8×9×5=360

Therefore, the LCM of 72, 90, and 120 is 360.