I am working on Integer factorization problem and I came to those two interesting expressions:
- $a,b,c,x$ are non negative integers
- $a,b < c$
$$\frac{ax + b}{c-x}$$ and $$\frac{ac + b}{c-x}$$
The interesting part about it, is that the division in both of the expressions provide the same reminder. I don't know if it happens for all the assignments or not, with the constants $a,b,c$ that I am working with it does! And my goal is to find such $x$ that will give me a reminder of 0.
Do you have any idea how can I use this knowledge about the reminder in order to achieve my goal?
Here are few examples where it happens.
- $\frac{59x + 100}{101 - x}$ has same reminder as $\frac{6059}{101 - x}$
- $\frac{74x + 21}{157 - x}$ has same reminder as $\frac{11639}{157 - x}$
One way to show that their remainders are the same is to show that their difference has no remainder: $$\frac{ax+b}{c-x}-\frac{ac+b}{c-x} = \frac{ax-ac}{c-x} = -a\frac{c-x}{c-x} = -a$$