Question

We have positive integers $a,$ $b,$ and $c$ such that $a > b > c.$ When $a,$ $b,$ and $c$ are divided by $19$, the remainders are $4,$ $2,$ and $18,$ respectively. When the number $2a b - c$ is divided by $19$, what is the remainder

Answer 1

We have for some integers
`i,j,k`,

`a \equiv 4 \pmod{19} \implies a = 19i + 4`

`b \equiv 2 \pmod{19} \implies b = 19j + 2`

`c \equiv 18 \pmod{19} \implies c = 19k + 18`

so that

`2a + b - c = 19(2i + j - k) - 8 \implies 2a+b-c \equiv -8 \equiv \boxed{11} \pmod{19}`