RSA signature verification with a different modulus

Question

I got a question on RSA signature verification. To verify, we calculate $y^e$ mod n (where y = $x^d$ mod n being the signature) to get x and compare with the original message.

Is it possible to verify the signature if we use another number n'= kn where k and n is relatively prime ?

Answer 1

No. If the signature has been created with $(n,d)$, it must be checked with $(n,e)$ with the same $n$.

(and in real life there'd be a hash and a padding scheme involved as well, making it even more unlikely that any other modulus could work).