<s>[INST] How many integers between $10000$ and $99999$ ( inclusive) are divisible by $3$ or $5$ or $7$? How many integers between $10000$ and $99999$ (inclusive) are divisible by $3$ or $5$ or $7$ ?

My Try :
Total Integers between $10000$ and $99999$ are $89999$.

$\left\lfloor\frac{89999}{3}\right\rfloor+\left\lfloor\frac{89999}{5}\right\rfloor+\left\lfloor\frac{89999}{7}\right\rfloor$ - $\left\lfloor\frac{89999}{3\times5}\right\rfloor-\left\lfloor\frac{89999}{3\times7}\right\rfloor-\left\lfloor\frac{89999}{5\times7}\right\rfloor$ + $\left\lfloor\frac{89999}{3\times5\times7}\right\rfloor$ =
$48857$

I don't have an answer for this. Am I right here ?
 [/INST] Another way:
$10000=3\times 3333+1\implies$ First integer $>10000$ divisible by $3$ is $10002$.
So if we count a number of integers in sequence $10002,10005,\cdots,99999$ we have our answer. the common difference is $3$ first term is $10002$ hence, $a_n=10002+(n-1)3=99999\implies n=30000$. Similarly for others too. 
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