Question

What are the common multiples of 9 and 15?

Answer 1

Hint: To find the common multiples of 9 and 15, we are going to do the prime factorization of 9 and 15 separately and then will find the L.C.M of both these numbers. L.C.M (Least Common Multiple) is found by finding the common factors from the two numbers and then multiplying the remaining factors with the common factor. Now, multiply the L.C.M with 1, 2, 3 and so on. Multiplying the L.C.M by 1, 2, 3… will give you the common multiples of 9 and 15.

Complete step-by-step answer:
In the above problem, we have given the following two numbers which we have to find the common multiples:
9 and 15
Now, we are going to find the prime factorization of 9 and 15.
First of all, we are going to find the prime factorization for 9 which is equal to:
$9=3\times 3$
After that, we are going to find the prime factorization for 15 which is equal to:
$15=3\times 5$
Now, comparing the prime factorization for the two numbers, the factors (or multiples) of 9 are (3, 3) and the factors (or multiples) of 15 are (3, 5) so you can see that 3 is the common factor (or multiple) among both the numbers. In the below, we are showing the common factor by underlining them.
$\begin{align}
  & 9=\underline{3}\times 3 \\
 & 15=\underline{3}\times 5 \\
\end{align}$
As you can see that common factor is 3 and uncommon factors from 9 and 15 are 3 and 5 so multiplying 3 by 3 and 5 we get the L.C.M as:
$3\times 3\times 5=45$
From the above, we got the L.C.M of 9 and 15 as 45. Now, the common multiples of 9 and 15 are found by multiplying 45 by 1, 2, 3 and so on.
Multiplying 45 by 1 we get,
$45\times 1=45$
Multiplying 45 by 2 we get,
$45\times 2=90$
Multiplying 45 by 3 we get,
$45\times 3=135$
Hence, the common multiples of 9 and 15 are 45, 90, 135….

Note: The other way of solving the above problem is to write the multiples of 9 and 15 separately and then find the common multiples among 5 and 19.
The multiples of 9 are as follows:
$9,18,27,36,45,54,63,72,81,90,99,108,117,126,135..$
The multiples of 15 are as follows:
$15,30,45,60,75,90,105,120,135...$
Now, mark the common multiples among the multiples listed:
$9,18,27,36,\underline{45},54,63,72,81,90,99,108,117,126,\underline{135}..$
$15,30,\underline{45},60,75,90,105,120,\underline{135}...$
From the above, you can see that 45 and 135 are the common multiples. In this way by writing further multiples of 9 and 15, we can find the common multiples of 9 and 15.