The number of 4 -digit numbers which are neither multiple of 7 nor...

The total possible $4$ digit numbers are $9 \times 10 \times 10 \times 10 = 9000$

Let, $X$ is the $4$ digit number divisible by $3$

$1002, 1005, \ldots \ldots \ldots , 9999$

$a = 1002, d = 3 \text{ and } {a}_{n} = 9999$

$9999 = 1002 + \left({n}_{3} - 1\right)3$

${n}_{3} = 3000$

Let, $Y$ is the $4$ digit numbers divisible by $7$

$1001, 1008, \ldots \ldots \ldots , 9996$

$a = 1001, d = 7 \text{ and } {a}_{n} = 9996$

$9996 = 1001 + \left({n}_{7} - 1\right)7$

${n}_{7} = 1286$

Now, $4$ digit numbers divisible by both $3$ and $7$

$1008, 1029, . . . . . . . . . . . 9996$

$a = 1008, d = 21, {a}_{n} = 9996$

Now, $9996 = 1008 + \left(n - 1\right)21$

$n = 429$

Hence, $4$ digit numbers neither divisible by $7$ nor $3$

$= 9000 - 1286 - 3000 + 429$

$= 5143$