JEE Main 2023 — Mathematics Algebra

Four-digit numbers that are less than or equal to $2800$2800 are ${1000,1001,1002,....,2800}$1000,1001,1002,....,2800

Numbers divisible by $3$3 are ${1002,1005,....,2796,2799}$1002,1005,....,2796,2799

Let total numbers be $n$n, then

$$1002+(n-1)3=2799$$1002+(n−1)3=2799

$$\Rightarrow n=600$$⇒n=600

Numbers divisible by $11$11 are ${1001,1012,...,2794}$1001,1012,...,2794

Let total numbers be $m$m, then

$$2794=1001+(m-1)11$$2794=1001+(m−1)11

$$\Rightarrow m=164$$⇒m=164

Numbers divisible by $33$33 are

$${1023,1056,...,2772}$$1023,1056,...,2772

Let total numbers be $r$r, then

$$2772=1023+(r-1)33$$2772=1023+(r−1)33

$$\Rightarrow r=54$$⇒r=54

Number of $4$4-digit numbers that are less than or equal to $2800$2800 and either divisible by $3$3 or by $11$11 , is equal to$=600+164-54=710$=600+164−54=710