The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is .

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

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

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

a=1002,d=3&{a}_{n}=9999

$$9999=1002+({n}_{3}-1)3$$9999=1002+(n3​−1)3

$${n}_{3}=3000$$n3​=3000

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

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

a=1001,d=7&{a}_{n}=9996

$$9996=1001+({n}_{7}-1)7$$9996=1001+(n7​−1)7

$${n}_{7}=1286$$n7​=1286

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

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

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

Now, $9996=1008+(n-1)21$9996=1008+(n−1)21

$$n=429$$n=429

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

$$=9000-1286-3000+429$$=9000−1286−3000+429

$$=5143$$=5143