Question

How many positive integers between $1000$ and $9999$ inclusive are divisible by either $5$ or $7$?

Answer 1

Finding the number of positive integers between $1000$ and $9999$ inclusive that are divisible by either $5$ or $7$.

Total number of positive integers between $1000$ and $9999$ inclusive that are $=9999-999$

$=9000$

The total number of positive integers between $1000$ and $9999$ inclusive that are divisible by $5$ can be calculated by dividing the total number of positive integers by $5$.

$\begin{array}{rcl}\mathrm{number}\mathrm{of}\mathrm{positive}\mathrm{integers}& =& \frac{9000}{5}\\ & =& 1800\end{array}$

The total number of positive integers between $1000$ and $9999$ inclusive that are divisible by $7$ can be calculated by dividing the total number of positive integers by $7$.

$\begin{array}{rcl}\mathrm{number}\mathrm{of}\mathrm{positive}\mathrm{integers}& =& \frac{9000}{7}\\ & =& 1285.7\\ & \approx & 1286\end{array}$

Total number of positive integers between $1000$ and $9999$ inclusive that are divisible by both $5$ and $7$ can be calculated by dividing the total number of positive integers by $5\times 7$.

$\begin{array}{rcl}\mathrm{number}\mathrm{of}\mathrm{positive}\mathrm{integers}& =& \frac{9000}{35}\\ & =& 257\end{array}$

So, the total number of positive integers between $1000$ and $9999$ inclusive that are divisible by either $5$ or $7$ can be calculated:

$\begin{array}{rcl}\mathrm{number}\mathrm{of}\mathrm{positive}\mathrm{integers}& =& 1800+1286-257\\ & =& 2829\end{array}$

Hence, the number of positive integers between $1000$ and $9999$ inclusive that are divisible by either $5$ or $7$ are $2829$.