Question

What is the least common multiple of 3, 9, and 15?

Answer 1

Hint: We have to find the least common multiple of 3, 9, and 15, which is abbreviated as LCM. So, we will first find the factors of 3, 9 and 15 using the common division method. Then, we will note down common factors and multiply them to get the LCM.

Complete step by step solution:
We know that the least common multiple of any two numbers is the least number which is exactly divisible by both the numbers. The least common multiple is mostly abbreviated as LCM. We can find the LCM of the given numbers by the help of just listing the multiples or by common division method.
We have to find LCMs of 3, 9 and 15.
We will use the common division method. In this method we divide the numbers by least prime number together and stop it when the further division of these numbers is not possible with the help of prime numbers and (1,1,1) left at the end. Then we just multiply the divisors and this will give the least common multiple of the given numbers
LCM of 3, 9 and 15 is,
$\begin{align}
  & 3\left| \!{\underline {\,
  3,15,9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  1,5,3\ \ \,}} \right. \\
 & 5\left| \!{\underline {\,
  1,5,1\ \ \ \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,1,1\quad \,}} \right. \\
\end{align}$
Thus, the L.C.M of 3, 9 and 15 is
$\Rightarrow L.C.M=3\times 3\times 5$
$\Rightarrow L.C.M=45$
Hence, the L.C.M of 3, 9 and 15 is 45.

Note: Students should also know another method prime factorization, here we will write the given numbers in terms of prime factors, if a prime factor will be repeating then we will write them in exponential form. Now, the LCM will be the product of the prime factors along with their highest exponent present. That is
$\begin{align}
  & \Rightarrow 3={{3}^{1}} \\
 & \Rightarrow 9={{3}^{2}} \\
 & \Rightarrow 15={{3}^{1}}\times {{5}^{1}} \\
\end{align}$
Therefore, the LCM is the product of the prime factors along with their highest exponent present that is
$\Rightarrow L.C.M={{3}^{2}}\times {{5}^{1}}=45$