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  4. Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to

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Q. Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to

JEE MainJEE Main 2023Permutations and Combinations

Solution:

$1000-2799$
Divisible by 3
$1002+(n-1) 3=2799$
$n=600$
Divisible by 11
$ 1-2799 \rightarrow\left[\frac{2799}{11}\right]=[254]=254$
$ 1-999=\left[\frac{999}{11}\right]=90 $
$ 1000-2799=254-90=164$
Divisible by 33
$ 1-2799 \rightarrow\left[\frac{2799}{33}\right]=84 $
$ 1-999 \rightarrow\left[\frac{999}{33}\right]=30$
$ 1000-2799 \rightarrow 54$
$ \therefore n (3)+ n (11)- n (33) $
$ 600+164-54=710$