number theory grade9 thailand

Question

Let $x, y$ be positive integers such that $21x^2​+16xy+3y^2​=324, 000$ what is the probability that $x$ and $y$ are positive even integers.

my attempt :$(3x+y)(7x+3y)=324, 000$ and i try to factor $324,000$ but $324,000$ has $120$ factors and i don't know how to do next

Answer 1

Hint: Note that $3x+y$ and $7x+3y$ are of the same parity, which must be even, so $x$ and $y$ have the same parity. Consider $ab=324,000$ where $a,b$ are both even. Find $x,y$ as a function of $a,b$. Only some of the $a,b$ pairs will have $x,y$ both positive. $x$ and $y$ will be even when both $a,b$ have more than one factor of $2$.

Answer 2

Write $$21x^2+16xy+3y^2=32400$$ in closed form explicitly $$(3x+y)(7x+3y)=32400.$$ Then solve for $$7x^2+\frac{16xy}{3}+y^2=10800.$$

Subtract $7x^2$ on both sides, complete the squares. We get

$$y=\sqrt{\frac{x^2}{9}}+108000-\frac{8x}{3}.$$

$(x,y)=(-40494,94482)$.