The number of numbers lying between 100 and 500  that are divisible by 7 but not by 21 is

The numbers between 100 and 500 that are divisible  by 7 are 105, 112, 119, 126, 133, 140, 147, …, 483, 490, 497.

Let such numbers be n.

Then,$497=105+(\mathrm{n}-1)\times 7;\text{ or }\mathrm{n}=57$.

The numbers between 100 and 500 that are divisible by 21 are 105, 126, 147, …, 483.Let such numbers be m.
Then 483 = 105 + (m – 1) × 21; or m = 19.
Hence, the required number = n – m = 57 – 19 = 38.