JEE Main 2021 (Online) 26th February Evening ShiftQuestion 1
The total number of $4$-digit numbers whose greatest common divisor with $18$ is $3$, is $\_\_\_\_\_\_$ .
Permutation and Combination
Answer$1000$
Solution
$\gcd(n,18)=3$ iff $3\mid n$, $n$ odd, and $9\nmid n$ (i.e. $n$ is an odd multiple of 3 not divisible by 9). Odd multiples of 3 are multiples of 3 with $n\equiv3\pmod6$; there are 1500 four-digit ones; multiples of 9 that are odd (odd multiples of 9): 500. $1500-500=1000$.