Question

What is the least common multiple of $16$ and $30$?

Answer 1

Hint: We will discuss Least Common Multiple in detail while solving this problem and then solve this problem by those definitions and examples. We will also use the prime factorization method also to solve this problem. Prime numbers are the numbers which have only two divisors.

Complete answer:
In mathematics, Least Common Multiple (LCM) of two numbers is defined as the least among the common multiples of those two numbers. It can be calculated for two or more numbers.
Let us now know the method of finding LCM of two numbers.
Let the two numbers be $16$ and $30$.
The multiples of 16 are: $16,32,48,64,80,96,112,128,144,160,176,192,208,224,240,....$
The multiples of 30 are: $30,60,90,120,150,180,210,240,....$
When you observe, the multiples that are common for $16$ and $30$are: $240,480....$
So, the least one among the common multiples is $240$.
So, Least Common Multiple of $16$ and $30$ is $240$.
LCM can also be calculated in this way.
\[\begin{align}
  & 2\left| \!{\underline {\,
  16,30\,}} \right. \\
 & 2\left| \!{\underline {\,
  8,15\,}} \right. \\
 & 2\left| \!{\underline {\,
  4,15\,}} \right. \\
 & 2\left| \!{\underline {\,
  1,15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  1,3 \,}} \right. \\
 & 3\left| \!{\underline {\,
  1,1 \,}} \right. \\
\end{align}\]
This is the prime factorization method.
Now, we have to calculate the LCM of 16 and 30.
So, \[LCM = 2 \times 2 \times 2 \times 2 \times 5 \times 3\]
\[ \Rightarrow LCM = 240\]
So, LCM of 16 and 30 is 240.

Note:
We have to consider the least value among the multiples that we got in common though we got many multiples in common. We can try both methods as per our convenience. But if we have more than two numbers for which we need to find LCM, then the best method is the prime factorization method.
The LCM of two co-primes is equal to the product of those two numbers. Co-primes are the numbers which have only 1 as a common divisor.
For example, LCM of 3 and 4 is 12, which is a product of 3 and 4.