Question

Find the L.C.M: - 36, 60, 72.

Answer 1

Hint: Write the given numbers as the product of their prime factors. Now, take the prime factors having the highest exponent and multiply them together to get the required L.C.M of the given numbers.

Complete step-by-step solution
Here, we have to find the L.C.M of these given numbers 36, 60, and 72. First, let us know about L.C.M.
In arithmetic and number theory, the least common multiple of two integers x and y is the smallest positive integer that is divisible by both x and y. Here, x and y must not be 0. There are many methods to determine the L.C.M of two or more given numbers. Here, we will use the method of prime factorization.
In the method of prime factorization, we write the given numbers as the product of their prime factors. Now, the L.C.M will be the product of multiplying the highest power of each prime number together.
Now, let us come to the question. Here, we have three numbers 36, 60 and 72. So, writing them as the product of their primes, we get,
\[\begin{align}
  & \Rightarrow 36=2\times 2\times 3\times 3={{2}^{2}}\times {{3}^{2}} \\
 & \Rightarrow 60=2\times 2\times 3\times 5={{2}^{2}}\times {{3}^{1}}\times {{5}^{1}} \\
 & \Rightarrow 72=2\times 2\times 2\times 3\times 3={{2}^{3}}\times {{3}^{2}} \\
\end{align}\]
Clearly, we can see that the highest power of three prime factors 2, 3 and 5 are \[{{2}^{3}},{{3}^{2}}\] and \[{{5}^{1}}\] respectively. Therefore, taking their product, we get,
\[\Rightarrow \] L.C.M = \[{{2}^{3}}\times {{3}^{2}}\times {{5}^{1}}\]
\[\Rightarrow \] L.C.M = \[8\times 9\times 5\]
\[\Rightarrow \] L.C.M = 360
Hence, the L.C.M of 36, 60 and 72 is 360.

Note: We must not get confused in the process of finding the H.C.F and the L.C.M. In the process of finding H.C.F, we consider only common prime factors that are present in the prime factorization of given numbers. You may know that we can also write the multiples of the given numbers and then check the first common multiple that will appear to find the L.C.M, but here the numbers are large so it is advised to use the method of prime factorization only.