Question

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

Answer 1

$A=4-$ digit numbers divisible by 3
$A=1002,1005, \ldots, 9999$
$9999=1002+(n-1) 3$
$\Rightarrow ( n -1) 3=8997 \Rightarrow n =3000$
$B =4-$ digit numbers divisible by $ 7 $
$ B =1001,1008, \ldots, 9996 $
$\Rightarrow 9996=1001+( n -1) 7 $
$\Rightarrow n =1286 $
$ A \cap B =1008,1029, \ldots, 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 $=9000-3857=5143$