Find the smallest number such that
(i) it leaves a remainder 2 when divided by 3 ;
(ii) it leaves a remainder 3 when divided by 5 ;
(iii) it leaves a remainder 5 when divided by 7 .
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.