[Math] Find the value of $2xy$ .

algebra-precalculuslinear-diophantine-equationsnumber theory

If $13x+17y=643$ ,$\{x,y\}\in \mathbb{N}$, then what is the value of two times the product of
$x$ and $y$ ?

Options

$a.)\ 744\quad \quad \quad \quad \quad
b.)\ 844\\
\color{green}{c.)\ 924}\quad \quad \quad \quad \quad
d.)\ 884\\$

I tried,

$13x+17y \pmod{13}\equiv 0\\
\implies 2y \pmod{13}\equiv 3 \\
\implies y=8
\implies y=8, x=39$

$2xy=624$

I look for a short and simple way .

I have studied maths up to $12$th grade.

Best Answer

When applying mod $13$, the equation $13x+17y=643$ becomes $$4y\equiv 6\pmod{13}$$ or $$40y\equiv 60\pmod {13}$$ that is, $y\equiv 8\pmod {13}$.

Now, to find $x$, apply mod $17$:

$$13x\equiv 14\pmod{17}$$ or $$4\cdot 13x\equiv 56\pmod {17}$$ thus, $x\equiv 5\pmod{17}$.

Now we are to find the concrete values of $x$ and $y$: $$13(17u+5)+17(13v+8)=643$$ which yields $$221(u+v)+201=643$$ therefore, $u+v=2$. Since $u$ and $v$ must not be negative, we have three possibilities:

  • $u=2$, $v=0$. Then $x=39$, $y=8$, so $2xy=624$.
  • $u=v=1$. Then $x=22$, $y=21$. Then $2xy=924$.
  • $u=0$, $v=2$. Then $x=5$, $y=34$. Then $2xy=340$.
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