- Home
- Standard 11
- Mathematics
ધારોકે $A=\{n \in[100,700] \cap N: n$ એ $3$ નો ગુણિત પણ નથી કે $4$ નો ગુણિત પણ નથી $\}$. તો $A$ ના ધટકોની સંખ્યા ........... છે.
$300$
$280$
$310$
$290$
Solution
$ \mathrm{n}(3) \Rightarrow \text { multiple of } 3 $
$ 102,105,108, \ldots . ., 699 $
$ \mathrm{~T}_{\mathrm{n}}=699=102+(\mathrm{n}-1)(3) $
$ \mathrm{n}=200 $
$ \mathrm{n}(3)=200 $
$ \because \mathrm{n}(4) \Rightarrow$ multiple of $4$
$ 100,104,108, \ldots ., 700 $
$ T_n=700=100+(n-1)(4) $
$ n=151 $
$ n(4)=151 $
$ \mathrm{n}(3 \cap 4) \Rightarrow \text { multiple of } 3 \& 4 \text { both } $
$ 108,120,132, \ldots ., 696 $
$ T_n=696=108+(n-1)(12) $
$ \mathrm{n}=50 $
$ \mathrm{n}(3 \cap 4)=50 $
$ \mathrm{n}(3 \cup 4)=\mathrm{n}(3)+\mathrm{n}(4)-\mathrm{n}(3 \cap 4) $
$ \quad=200+151-50 $
$ =301$
$\mathrm{n}(\overline{3 \cup 4})=$ Total $-\mathrm{n}(3 \cup 4)=$ neither a multiple of $3$ nor a multiple of $4$
$=601-301=300$
