I am doing simple question on congruence modulo arithmetic, and my attempt is:
$77546 + x - 465 = 513* 663 \pmod{92}$
=> $77081 + x = (460+53)*(644+19) \pmod{92}$
=> $ (92000-14919) + x = (53)*(19) \pmod{92}$
=> $ (92000-(9200+5719)) + x = (-39)*(19) \pmod{92} $
=> $ (92000-(9200+4600+920+184+15))+ x =(-40+1)*(19) \pmod{92} $
=> $ -15 + x =(-760+19)\pmod{92} $
=> $ -15 + x =-741\pmod{92} $
=> $ -15 + x =(-736 -5)\pmod{92} $
=> $ 77+ x=-5 \pmod{92}$
=> $77+ x= 87 \pmod{92}$
=> $x = 10\pmod{92}.$
The minimum positive value (as need positive values only) is $x = 10$. It is correct, as per the answer.
But, I feel confused over the value $10 \pmod{92} $ that can be added. Can I add any value belonging to the set : $10+ 92n, \forall n \in \mathbb{Z+}$.
The next question is only about repeating the correct formatting for a new question, with nothing to ask:
$2241- 43*7572 \pmod{94}$
=> $(1880 + 376 -15) -43*(9400-1880 + 52) \pmod{94}$
=> $-15 -43*52 \pmod{94}$
=> $-15 -(40+3)*(50+2) \pmod{94}$
=> $-15 -(2000+80 +150 +6) \pmod{94}$
=> $-15 -(2000+236) \pmod{94}$
=> $-15 -(2000+(188+48)) \pmod{94}$
=> $-15 -2048 \pmod{94}$
=> $-15 -74 \pmod{94}$
=> $-89 \pmod{94}$
=> $5 \pmod{94}$