chinese reminder theorem

Question

"A house wife spends 1770 Rupees for purchasing mangoes and apples. She pays 31 Rupees for each mango and 21 Rupees for each apple. If she buys more mangoes than apples, how many mangoes and apples she buys"

I know how to solve this using dolphins equation. can it be solved using chinese reminder theorem. this is what i think. $1770 \equiv x $ (mod $ 31)$ $1770 \equiv y $ (mod $ 21)$. I know how to do the chinese reminder theorem for 1 variable. please help me with this.

Answer 1

$1770=31M+21A$, so $3\equiv 21A\pmod{31}$.

To solve that congruence, you could use the Euclidean algorithm:

$31=1\times21+10$

$21=2\times10+1$

So $1=21-2\times10=21-2\times(31-21)=3\times21-2\times31$,

so $3\equiv9\times21\pmod{31}$.

Can you take it from here?

Answer 2

$1770 = 31m\! +\! 21a\,\Rightarrow\, \color{#c00}{\bmod 21\!:\ m \equiv} \dfrac{1770}{31}\equiv \dfrac{1770}{10}\equiv 177\equiv \color{#c00}9$

Therefore $\, \color{#c00}{m = 9\!+\!21}k\, \Rightarrow\ a = (1770\!-\!31m)/21 = 71\!-\!31k$