Attacking RSA

Suppose Nikita's public key is $(n,e)$ and her decryption key is $d$ , so $ed\equiv 1\pmod{\varphi (n)}$ . If somehow we compute the factorization $n=pq$ , then we can compute $\varphi (n)=(p-1)(q-1)$ and hence compute $d$ . Thus if we can factor $n$ then we can break the corresponding RSA public-key cryptosystem.