Factoring $n$ Given $\varphi (n)$

If you know $\varphi (n)$ then it is easy to factor $n$:
Suppose $n=pq$. Given $\varphi (n)$, it is very easy to compute $p$ and $q$. We have

$$\varphi (n) = (p-1)(q-1) = pq-(p+q)+1,$$

so we know both $pq=n$ and $p+q = n+1 - \varphi (n)$. Thus we know the polynomial

$$x^2 - (p+q)x + pq = (x-p)(x-q)$$

whose roots are $p$ and $q$. These roots can be found using the quadratic formula.

Example 1.1  

? n=nextprime(random(10^10))*nextprime(random(10^10));
? phin=eulerphi(n);
? f = x^2 - (n+1-phin)*x + n
%6 = x^2 - 12422732288*x + 31615577110997599711
? polroots(f)
%7 = [3572144239, 8850588049]
? n
%8 = 31615577110997599711
? 3572144239*8850588049
%9 = 31615577110997599711