Question

Find the LCM of $216$ and $240$.

Answer 1

Hint: Write as the product of prime factors of each number and then choose the common and uncommon prime factors with the greatest exponent. Multiply the common and uncommon prime factors with the greatest exponents.

Complete step-by-step answer:
We have to find the LCM of $216$ and $240$.
We know that LCM of two given numbers is the smallest number which is divisible by both the given numbers.
Writing $216$ as the product of prime factors we get $216=2\times 2\times 2\times 3\times 3\times 3$.
Writing $240$ as the product of prime factors we get $240=2\times 2\times 2\times 2\times 3\times 5$.
Further simplifying we can write $216$ and $240$ as $216={{2}^{3}}\times {{3}^{3}}$ and $240={{2}^{4}}\times 3\times 5$
Choose the common and uncommon prime factors with the greatest exponent.
Common prime factors are $2,3$.
Common prime factors with the greatest exponent \[{{2}^{4}},{{3}^{3}}\].
Uncommon prime factor is \[5\].
Uncommon prime factor with the greatest exponent is \[{{5}^{1}}\].
To get the LCM multiply the common and uncommon prime factors with the greatest exponents of the numbers.
Hence LCM of $216$ and $240$ can be obtained by ${{2}^{4}}\times {{3}^{3}}\times 5$.
Now simplifying ${{2}^{4}}\times {{3}^{3}}\times 5$ we get ${{2}^{4}}\times {{3}^{3}}\times 5=2160$.
Also we can see that $2160$is the least number which is completely divisible by both the given numbers $216$ and $240$.
Therefore the LCM of $216$ and $240$ is $2160$.
This is the required solution of the given problem.

Note: In this problem students need to take care of finding the product of prime factors of the numbers and the exponent of common and uncommon prime factors. Students can solve using the division method to find the LCM.