Use the Chinese Remainder Theorem to give the general solution to the equations: x = 9 (mod 10) x = 5 (mod 11) x = 9 (mod 13)

Question

Use the Chinese Remainder Theorem to give the general solution to the equations:

x = 9 (mod 10) x = 5 (mod 11) x = 9 (mod 13)

I got x= 269 (mod 1430) is this wrong?

Answer 1

Correct. $\,10,13\mid x\!-\!9\,\iff 10\cdot 13\mid x\!-\!9\iff x = 9 + 130k.$

${\rm mod}\ 11\!:\,\ 5\equiv x\equiv 9+130k\equiv 9-2k$ $\iff 2k\equiv 4\iff {k\equiv 2}\iff \color{#c00}{k= 2\!+\!11n}$

Therefore $\,\ x = 9+130\,\color{#c00}k = 9+130(\color{#c00}{2\!+\!11n}) = 269+ 130(11n)$