Question

How many positive integers between $1000$ and $9999$ inclusive are divisible by nine?

Answer 1

Finding the numbers $\left(n\right)$ between $1000$ and $9999$ inclusive that are divisible by $9$:

• To get the number divisible by $9$, there must be a difference of $9$ between each number.
• Using the Arithmetic Progression formula $l=a+(n-1)d$:

Putting the value of the first digit $\left(a\right)$ as $1000$, the last digit $\left(l\right)$ is $9999$ and difference $\left(d\right)$ as $9$ in the above formula:

$\begin{array}{rcl}9999& =& 1000+(n-1)9\\ & \Rightarrow & n-1=999.88\\ & \Rightarrow & n=1000.88\\ & \Rightarrow & n\approx 1000\end{array}$

Hence, $1000$ positive integers between $1000$ and $9999$ inclusive are divisible by nine.