Question

How many positive integers between $1000$ and $9999$ inclusive are not divisible by either $5$ or $7$?

Answer 1

Finding the positive integers between $1000$ and $9999$ inclusive that are not divisible by either $5$ or $7$:

• The total number of positive integers between $1000$ and $9999$ inclusive are $9999-999=9000$.
• The total number of the 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 the positive integers between $1000$ and $9999$ inclusive that are divisible by $7$ can be calculated by dividing the total number of positive integers by $7$.

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

The total number of the positive integers between $1000$ and $9999$ inclusive that are divisible by $5$ and $7$ can be calculated by dividing the total number of positive integers by $35$.

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

Hence, the total number of the positive integers between $1000$ and $9999$ inclusive that are not divisible by either $5$ or $7$ can be calculated as follows:

$\begin{array}{rcl}\mathrm{number}\mathrm{of}\mathrm{positive}\mathrm{integers}& =& 9000-1800-1286+257\\ & =& 6171\end{array}$

Hence, there are $6171$ positive integers between $1000$ and $9999$ inclusive that are not divisible by either $5$ or $7$.