joe123x
what is the proof from the equation that the remainder is always less than the divisor?
eq: m/x = q + 7/x
The remainder should always be less than the divisor because that is how we define the remainder in the context of division:
If \(x\) and \(y\) are positive integers, there exist unique integers \(q\) and \(r\), called the quotient and remainder, respectively, such that \(y =divisor*quotient+remainder= xq + r\) and \(0\leq{r}<x\).When we divide one number (the dividend) by another number (the divisor), we are essentially asking how many times the divisor can fit into the dividend. The quotient is the whole number answer to this question, and the remainder is the amount that is left over after we've taken out as many whole divisors as we can.
For example, if we divide 13 by 3, we get a quotient of 4 and a remainder of 1. This means that 3 times 4 can fit into 13, with 1 left over. The remainder is the "leftover" amount, which must be less than the divisor (in this case, less than 3) because if it were equal to or greater than the divisor, then we could fit another whole divisor into the dividend.
Therefore, the remainder must always be less than the divisor in order for the result of the division to make sense and be consistent with the way we define the operation.