Question

Find the number of integers between $100$ and $1000$ that are not divisible by $7$.

Answer 1

Find the number of integers between $100$ and $1000$ which are not divisible by $7$.

The numbers between $100$ and $1000$ which are exactly divisible by $7$.

$\Rightarrow$ $\frac{100}{7}=14.28<15$

$\Rightarrow$ $\frac{1000}{7}=142.85>142$

So, the numbers which are exactly divisible by $7$ are,

First number $=7\times 15$

$=105$

Second number $=7\times 16$

$=112$

Third number $=7\times 17$

$=119$

and so on till $142\times 7=994$

Also, $142-14=128$

There are total $899$ numbers between $100$ and $1000$ from which $128$ numbers are divisible by $7$.

So, the remaining numbers are,

$\therefore 899-128=771$

Therefore, the number of integers which are not divisible by $7$ is $771$.