Question

Given a number
a) that is divisible by 3, 9, and 5.
b) that is divisible by 3, 9, and 10.

Answer 1

Hint: We solve this problem by first realizing that the number divisible by the given numbers is the multiple common to the given numbers. Then we find the LCM of given numbers in each subpart of the question using the long division method. Then we give the answer as the LCM of the given numbers.

Complete step-by-step solution:
a) number divisible by 3, 9 and 5.
We need to find the number that is divisible by the numbers 3, 9, and 5, that is we need to find a number that is a multiple by 3, 9, and 5.
So, we need to find the number that is a common multiple of 3, 9, and 5.
So, first let us find the least common multiple, that is LCM of 3, 9, and 5.
$\begin{align}
  & 3\left| \!{\underline {\,
  3,\text{9,5} \,}} \right. \\
 & 3\left| \!{\underline {\,
  1,3\text{,5} \,}} \right. \\
 & 5\left| \!{\underline {\,
  1,1\text{,5} \,}} \right. \\
 & \text{ }1,1\text{,1} \\
\end{align}$
So, we get the LCM of 3, 9 and 5 as
$\begin{align}
  & \Rightarrow LCM=3\times 3\times 5\times 1\times 1\times 1 \\
 & \Rightarrow LCM=45 \\
\end{align}$
So, the LCM of 3, 9 and 5 is 45.
So, we get that 45 is divisible by 3, 9, and 5.
Hence answer is 45.
b) number divisible by 3, 9, and 10.
We need to find the number that is divisible by the numbers 3, 9, and 10, that is we need to find a number that is a multiple by 3, 9, and 10.
So, we need to find the number that is a common multiple of 3, 9, and 10.
So, first let us find the least common multiple, that is LCM of 3, 9 and 10.
\[\begin{align}
  & 3\left| \!{\underline {\,
  3,\text{9,10} \,}} \right. \\
 & 3\left| \!{\underline {\,
  1,3\text{,10} \,}} \right. \\
 & 5\left| \!{\underline {\,
  1,1\text{,10} \,}} \right. \\
 & 2\left| \!{\underline {\,
  1,1\text{,2} \,}} \right. \\
 & \text{ }1,1\text{,1} \\
\end{align}\]
So, we get the LCM of 3, 9 and 10 as
$\begin{align}
  & \Rightarrow LCM=3\times 3\times 5\times 2\times 1\times 1\times 1 \\
 & \Rightarrow LCM=90 \\
\end{align}$
So, the LCM of 3, 9 and 10 is 90.
So, we get that 90 is divisible by 3, 9, and 10.
Hence answer is 90.

Note: We can also give the answers to the given sub-parts differently as
a) We got above that the LCM of 3, 9, and 5 is 45.
45 is the least common multiple of 3, 9, and 5. So, any multiple of 45 is also divisible by 3, 9, and 5. So, we can give any multiple of 45 as an answer.
As we see, 450 is a multiple of 45. So, 450 is divisible by 3, 9, and 5.
Hence answer is 450.
b) We got above that the LCM of 3, 9, and 10 is 90.
90 is the least common multiple of 3, 9, and 10. So, any multiple of 90 is also divisible by 3, 9, and 10. So, we can give any multiple of 90 as an answer.
As we see, 450 is a multiple of 90. So, 450 is divisible by 3, 9, and 10.
Hence answer is 450.