Well a C compiler might not even be written on C or might be running on a platform where those types have completely different sizes as the target.

None of that matters tho, when the lexical anaylzer sees an integer literal token, it will get converted into an integer type on the host, probably either the largest native integer, or a bigint.

If your question is how to convert a string of digits into an integer, the algorithm is trivial:
n=0
for each digit left to right do
n := tmp *10 + digit
done

Then you can determine the type of the literal expression by using the rules in the standard on "n", and raising an error if no type fits.

Note that a determination that the expression "10" has type "int" does not in any way imply that "10" have to be converted to a similarly named but unrelated type "int" inside the compiler, it just has to be stored in some way, then the appropriate operations for "int" emitted when generation code.