Question

Find the common multiples of 3, 4 and 9.

Answer 1

Hint:
Here, we will find the common multiples of the numbers by using the least common multiples of the given number. A common multiple of a number is defined as a whole number that is a shared multiple of each set of the numbers.

Complete step by step solution:
We are given with the numbers 3, 4 and 9.
We will first find the Least common multiple of the given number.
\[\begin{array}{*{20}{l}}
  3 | & 3&4&9 \\
\hline
  3 | & 1&4&3 \\
\hline
  2 | & 1&4&1 \\
\hline
  2 | & 1&2&1 \\
\hline
  1 | & 1&1&1
\end{array}\]
So, the least common multiple of the given number is given by
\[LCM\left( {3,4,9} \right) = 3 \times 3 \times 2 \times 2\]
Multiplying the terms, we get
\[ \Rightarrow LCM\left( {3,4,9} \right) = 36\]
So, the least common multiple of the numbers 3, 4, 9 is 36.
Now, we will find the common multiples of the numbers 3, 4, 9
The common multiples of the numbers are the multiples of the least common multiple of the number.
So, the common multiples of the numbers is given by
\[36 \times 1 = 36\]
\[36 \times 2 = 72\]
\[36 \times 3 = 108\]

Therefore, the common multiples of the numbers of \[3,4,9\] is \[36,72,108,............\]

Note:
We know that a number can have an infinite number of multiples. Therefore, any two or more set two numbers can have an infinite number of common multiples. The smallest common multiple is the L.C.M of the given numbers.
We can also find the common multiples of the given numbers from the multiples of the numbers.
\[\left. \begin{array}{l}3 \times 1 = 3\\3 \times 2 = 6\\3 \times 3 = 9\\3 \times 4 = 12\\3 \times 5 = 15\\3 \times 6 = 18\\3 \times 7 = 21\\3 \times 8 = 24\\3 \times 9 = 27\\3 \times 10 = 30\\3 \times 11 = 33\\3 \times 12 = 36\\3 \times 13 = 39\\3 \times 14 = 42\\3 \times 15 = 45\end{array} \right|\] \[\left. \begin{array}{l}4 \times 1 = 4\\4 \times 2 = 8\\4 \times 3 = 12\\4 \times 4 = 16\\4 \times 5 = 20\\4 \times 6 = 24\\4 \times 7 = 28\\4 \times 8 = 32\\4 \times 9 = 36\\4 \times 10 = 40\\4 \times 11 = 44\\4 \times 12 = 48\\4 \times 13 = 52\\4 \times 14 = 56\\4 \times 15 = 60\end{array} \right|\] \[\begin{array}{l}9 \times 1 = 9\\9 \times 2 = 18\\9 \times 3 = 27\\9 \times 4 = 36\\9 \times 5 = 45\\9 \times 6 = 54\\9 \times 7 = 63\\9 \times 8 = 72\\9 \times 9 = 81\\9 \times 10 = 90\\9 \times 11 = 99\\9 \times 12 = 108\\9 \times 13 = 117\\9 \times 14 = 126\\9 \times 15 = 135\end{array}\]
Here, from the multiplication table, we can see that only 36 is common to all 3 numbers till the 15th multiple.
Therefore, we can say that the common multiples of the numbers 3, 4, 9 is 36.