MHB Is It Possible to Find a Mod m and a Div m for Given Values of a and m?

[shamieh]

Find a div m and a mod m when

$a = -111$, $m = 99$

so I got the first one:
$a =-2*99+87$ => amodm=87 and adivm = -2

so if I had $a=−9999$ and $m = 101$ would that mean that $a = −2∗101-9797$ ? so a mod m $= -9797$ and a div m =$ -2$ ?

or should it be $a =-1 *101 + -9898$ => amodm = $-9898$ and adivm = $-1$

 

[Nono713]

The remainder (mod) is always between $0$ and $m - 1$ inclusive (for positive $m$), that's how the quotient is unique.

 

[shamieh]

The remainder (mod) is always between $0$ and $m - 1$ inclusive (for positive $m$), that's how the quotient is unique.

so the remainder mod, in this case is between $0$ and $100$ right (101-1)? This fact still doesn't answer my question though. I need to know if my method is correct or incorrect.

 

[Nono713]

so the remainder mod, in this case is between $0$ and $100$ right (101-1)? This fact still doesn't answer my question though. I need to know if my method is correct or incorrect.

Yes, and it does, your method is incorrect, for the remainder you found in the second case isn't in this interval and so is wrong.

In theory, to find the quotient and remainder for positive $m$ what you do is find the smallest nonnegative integer $r$ such that $a - r$ is a multiple of $m$ ($r$ will be the remainder) and then divide $a - r$ by $m$ to find $q$. In practice you usually want to find the quotient first by dividing $a$ by $m$ and rounding down, and then working out the remainder from that.

For instance with $a = -111$ and $m = 99$, you compute $a / m \approx -1.12$, which when rounded down gives $q= -2$, so that you have $a = qm + r$, that is, $-111 = (-2) 99 + r$, so $r = -111 - (-2)99 = 87$.

For $a = -9999$ and $m = 101$ we get $a / m = -99$ (it happens to be exact), so there's no need to round down and so $q = -99$, and we get $-9999 = (-99)101 + r$, so $r = -9999 - (-99)101 = 0$ (in fact you already knew the remainder was zero, since $101$ happened to divide $-9999$ as the division gave an integer).

 

[shamieh]

Thank you for the help bacterius. This is going to sound like a really dumb question but you are saying that I need to round down $-1.12$ to $-2$ ... How in the world does $-1.12$ round down to $-2$ ?

 

[shamieh]

OH! Bacterius I'm an idiot. I see what you are talking about. $\lfloor -1.12 \rfloor = -2$