A number that can be divided by both 18 and 9 must be a multiple of the least common multiple (LCM) of these two numbers.
First, we can find the LCM of 18 and 9:
- The prime factorization of 18 is \(2 \times 3^2\).
- The prime factorization of 9 is \(3^2\).
To find the LCM, we take the highest power of each prime factor:
- The maximum power of 2 is \(2^1\).
- The maximum power of 3 is \(3^2\).
Thus, the LCM of 18 and 9 is: \[ LCM(18, 9) = 2^1 \times 3^2 = 2 \times 9 = 18. \]
Therefore, a number that can be divided by both 18 and 9 must be a multiple of 18. This includes numbers like 18, 36, 54, 72, and so on. In general, any number of the form \(18n\), where \(n\) is a non-negative integer, will satisfy this condition.