Solve $10x + y =_3 x \wedge 10x+y =_5 y$
Task comes from concrete mathematics.
This is my approach
\begin{cases} 10x + y =_3 x \\ 10x+y =_5 y \end{cases}
\begin{cases} 50x + 5y =_{15} 5x \\ 30x+3y =_{15} 3y \end{cases} $$ 80x+8y =_{15} 5x + 3y $$ but I stucked there... how can it be continued?
in book they say that
We want $10x+6y =_{15} 10x+y$ so...
but I don't know how they got this equality.
If $10x+y\equiv x\pmod 3$ then $y\equiv0\pmod 3$, so $3$ divides $y$ and therefore $5y$.
$5$ also divides $5y$. Therefore $15$ divides $5y,$ so $10x+6y\equiv10x+y\pmod{15}$.