S = {1, 2, 3, .......... 2022}

HCF (n, 2022) = 1

$$\Rightarrow$$ n and 2022 have no common factor

Total elements = 2022

2022 = 2 $$\times$$ 3 $$\times$$ 337

M : numbers divisible by 2.

{2, 4, 6, ........, 2022}$$\,\,\,\,$$ n(M) = 1011

N : numbers divisible by 3.

{3, 6, 9, ........, 2022}$$\,\,\,\,$$ n(N) = 674

L : numbers divisible by 6.

{6, 12, 18, ........, 2022}$$\,\,\,\,$$ n(L) = 337

n(M $$\cup$$ N) = n(M) + n(N) $$-$$ n(L)

= 1011 + 674 $$-$$ 337

= 1348

0 = Number divisible by 337 but not in M $$\cup$$ N

{337, 1685}

Number divisible by 2, 3 or 337

= 1348 + 2 = 1350

Required probability $$ = {{2022 - 1350} \over {2022}}$$

$$ = {{672} \over {2022}}$$

$$ = {{112} \over {337}}$$