QJEEM23Ma1fa1: JEE Main 2023 Mathematics - Class 11 JEEM23Ma1fa1 Solution: The number of 3-digit numbers, that are divisible by either 2…

Step-by-Step Verified Solution

$A=$ Numbers divisible by 2

$B=$ Numbers divisible by 3

$C=$ Numbers divisible by 7

$n(A \cup B)=n(A)+n(B)-n(A \cap B)$

$=n(2)+n(3)-n(6)$

$n(A)=n(2)=100,102 \ldots ., 998=450$

$n(B)=n(3)=102,105, \ldots ., 999=30$

$n(A \cap B)=n(6)=102,108, \ldots ., 996=150$

$n(2$ or 3 $)=450+300-150=600$

Now,

$n(\mathrm{~A} \cap C)=n(14)=112,126, \ldots ., 994=64$

$n(A \cap B \cap C)=n(42)=126,168, \ldots ., 966=21$

$n(B \cap C)=n(21)=105,126, \ldots \ldots, 987,=43$

$n(2$ or 3 not by 7 $)=600-[64+43-21]$

$=514$