Question

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

Answer 1

Answer: 5143

$A=4-$ digit numbers divisible by 3
$A=1002,1005,\dots ,9999$
$9999=1002+(n-1)3$
$\Rightarrow (n-1)3=8997\Rightarrow n=3000$
$B=4-$ digit numbers divisible by $7$
$B=1001,1008,\dots ,9996$
$\Rightarrow 9996=1001+(n-1)7$
$\Rightarrow n=1286$
$A\cap B=1008,1029,\dots ,9996$
$9996=1008+(n-1)21$
$\Rightarrow n=429$
So, no divisible by either 3 or 7
$=3000+1286-429=3857$
total 4-digits numbers $=9000$
required numbers $=9000-3857=5143$