Number Theory

By trial and error find the smallest number to satisfy (i) and (ii) is 8.

Then go on by trialing numbers of 8 + 15n since 15 is the least common multiple of 3 and 5. 

You will find the answer to be 68 within just a few tries.

There should be a better and more official way by modulo arithmetic though so it would be great if someone can also add that to this thread.

We have:

$x = 2 (\mod 3)$

$x = 3( \mod 5)$

$x = 5 ( \mod 7)$

Using the Chinese Remainder Theorem, we have $x = 68 ( \mod 105)$, so the smallest value that works is $\color{brown}\boxed{68}$