Let $S=\{1,2,3, \ldots, 2022\}$. Then the probability, that a randomly chosen number n from the set S such that $\mathrm{HCF}\,(\mathrm{n}, 2022)=1$, is :

<p>S = {1, 2, 3, .......... 2022}</p> <p>HCF (n, 2022) = 1</p> <p>$\Rightarrow$ n and 2022 have no common factor</p> <p>Total elements = 2022</p> <p>2022 = 2 $\times$ 3 $\times$ 337</p> <p>M : numbers divisible by 2.</p> <p>{2, 4, 6, ........, 2022}$\,\,\,\,$ n(M) = 1011</p> <p>N : numbers divisible by 3.</p> <p>{3, 6, 9, ........, 2022}$\,\,\,\,$ n(N) = 674</p> <p>L : numbers divisible by 6.</p> <p>{6, 12, 18, ........, 2022}$\,\,\,\,$ n(L) = 337</p> <p>n(M $\cup$ N) = n(M) + n(N) $-$ n(L)</p> <p>= 1011 + 674 $-$ 337</p> <p>= 1348</p> <p>0 = Number divisible by 337 but not in M $\cup$ N</p> <p>{337, 1685}</p> <p>Number divisible by 2, 3 or 337</p> <p>= 1348 + 2 = 1350</p> <p>Required probability $= {{2022 - 1350} \over {2022}}$</p> <p>$= {{672} \over {2022}}$</p> <p>$= {{112} \over {337}}$</p>