In long division we have four parts to the overall expression.
Dividend => the number being divided
Divisor => the number being divided by
Quotient => the whole number result
Remainder => the non-divisible amount
3 from your example…
11 / 3 => 3.66..
but when we take
0.66.. and resolve it to a rational number (all repeating decimals are rational numbers) we get
2/3 by the divisor,
3, and we get
2. Thus, 11 modulo 3 is 2.
11 % 3 => 2
Now consider the whole number quotient,
3 * 3 => 9
11 - 9 => 2
Again, remainder of 2.
We can use the remainder as a signal of non-divisibility. If the remainder is not zero, then the dividend cannot be divided evenly by the divisor. We now know with certainty that 3 does not divide into 11. There will always be a remainder.
From what we’ve covered above we have a simple formula…
D - qd = R