How to find the factors of the modulus?

In the article "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems", the original RSA article, it is mentioned that Miller has shown that n (the modulus) can be factored using any multiple of φ(n).

Imagine I know the public and the private key. But what I really want is the factors of n, the p and q but I connot use any factorization algorithm in a large number of n.

In the Miller's article it is suppose to say how I can find the two factores, knowing the public and private key. But I cannot understand how exactly it is done. Does someone know? Or have a small example?

Appreciated