Find the largest possible integer n such that $\sqrt{n}+\sqrt{n+60}=m$ for some non-square integer m. The solution to this problem says that: 
Squaring both sides gives us that $n(n+60)$ is a perfect square
I do not understand this, can someone explain me why this is true.
squareing gives:
$2n+60+2\sqrt{n}\sqrt{n+60}=m^2$
embaresed that i missed it 
$\sqrt{n}\sqrt{n+60}$ must be equal to some positive integer which implies that
$n(n+60)$ must be equal to some squared integer 
 A: As you have found out it is necessary that $n(n+60)$ is a perfect square. Therefore we want $n(n+60)=(n+r)^2$, or $(60-2r)n=r^2$, for some integer $r\geq0$. It follows that $r$ has to be even, and writing $r=2s$ we arrive at the condition $$(15-s)n=s^2\tag{1}$$ for  integer $s$ and $n\geq0$. Considering the cases $0\leq s\leq 15$ in turn we see that the following pairs $(s,n)$ solve $(1)$:
$$(0,0),\quad(6,4),\quad(10,20),\quad(12,48),\quad(14,196)\ .$$
The largest occuring $n$ in this list is $196$, and this $n$ indeed leads to an integer $m=\sqrt{196}+\sqrt{196+60}=30$. As $30$ is not a square the answer to the question is $196$.
A: Obviously both $n$ and $n+60$ should be  squares so its product $n(n+60)$ is a square. On the other hand, $n$ cannot be greater than some integer because if $n=x^2$ and $n+60=y^2$ then making $y=x+h$ we must have $2xh+h^2= 60$ which clearly put a bound for $x$.
This bound is easy to calculate and one has $x\lt 30$ because $30^2+60\lt 31^2$ hence $30^2+60\lt 31^2$ cannot be a square.
Furthermore $$\begin{cases}n=x^2\\n+60=y^2\end{cases}\Rightarrow(y+x)(y-x)=60=2^2\cdot3\cdot5\Rightarrow(x,y)=(14,16)$$ this largest $(14,16)$ corresponding to the factorization $30\cdot2$ (the factorization $60\cdot1$ does not give integer solution).
Thus the largest possible $n$ is $\color{red}{n=14^2=196}$ which corresponds to the integer $m=30=14+16$
A: We need $n(n+60)$ to be a perfect square.  Note that
$$
n(n+60) = (n+30)^2-900
$$
and $900 = 449+451$, so we can have $n+30 = \frac{451}{2} = 226$, or $n = 196$, and $n(n+60) = 50176 = 224^2$.
