Idealized form

Number Theory Level 5

$\large{\color{#3D99F6}{\frac{xy}{x+y}}=\color{#D61F06}{2^{4}3^{5}5^{4}}}$

If $x$ and $y$ are positive integers, What is the number of ordered pairs of $(x,y)$ that satisfy the equation above?

1 solution

Sauparna Paul
Jul 26, 2015

The above equation can be reduced into two factors as

$xy+m^{2}=m(x+y)+m^{2}$
Adding $m^{2}$ on both sides.

$(x-m)(y-m)=m^{2}$

where $m=2^{4}3^{5}5^{4}$ . So, the number of ordered pair is same as the number of divisors of the $m^{2}$ i.e. $(8+1)(10+1)(8+1) =891$ .