I want to solve Linear congrunece : 9x+2 ≡ 6(mod 1453) using inverse of 9 mod 1453. Inverse of 9 mod 1453 is 323. Now to solve it I subtract 2 from left and right side which gives me 9x ≡ 4(mod 1453), therefore x = 161. 161 is an answer to equation 9x ≡ 4(mod 1453) , but answer is not correct to equation to 9x+2 ≡ 6(mod 1453). Can I state my answer then as (9*161)%1453 = 4 (or 9*161 ≡ 4(mod 1453)). Or did I mess up something during the calculating process.
Sorry for my English. It is not my first language.
From $9x\equiv 4\pmod{1453}$, we get $(323)(9)x\equiv (323)(4)\pmod{1453}$.
But $(323)(9)\equiv 1\pmod{1453}$, so we get $$x\equiv (323)(4)\pmod{1453}.$$ Thus $x\equiv 1292\pmod{1453}$.
Remark: We have $1292\equiv -161\pmod{1453}$, so alternately we could say that $x\equiv -161\pmod{1453}$.