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Q.
The number of numbers lying between $100$ and $500$ that are divisible by $7$ but not by $21$ is
Sequences and Series
Solution:
The numbers between $100$ and $500$ that are divisible by
$7$ are $105,112,119,126,133,140,147, \ldots, 483, 490,497 .$
Let such numbers be $n$.
Then, $497=105+(n-1) \times 7$;
or $n=57$.
The numbers between $100$ and $500$ that are divisible by
$21$ are $105,126,147, \ldots, 483$.
Let such numbers be $m$.
Then $483=105+(m-1) \times 21$;
or $m=19$.
Hence, the required number $=n-m$
$=57-19=38$.