What is N?

Since we have

$\begin{cases} N≡3\phantom{i}mod\phantom{i}7\\ N≡4\phantom{i}mod\phantom{i}5 \end{cases}$

By the Chinese Remainder Theorem, there is only one solution modulo $35$ , namely

$N≡24\phantom{i}mod\phantom{i}35$

And since $N$ is natural, the minimal $N$ is $\boxed{24}$

not quite tough to think.

at first taking $(7+3=10)$ which will create a remainder 3 when divided by 7. but if divided by 4 the remainder will be 2......[so, not applicable]

secondly,taking $(14+3=17)$ which will create the same impact for 7 but not for 4..................[so, not applicable]

then, taking $(21+3=24)$ ,this, time the remainder of 7 is 3 and remainder of 5 is 4.

so, $24$ is the smallest possible value.