REMAINDERS

I think it's best maybe to start with a simpler version of that formula: y = Qx + r

So, for instance, 25 = 2(11)+ 3

y=25, x=11, Q=2, and r = 3. 25/11 has remainder three.

So if we divide both sides by 11:

25/11 = 2 + 3/11

Take the general formula y = Qx + r and divide by x, and you'll get y/x = Q + r/x

We typically say the remainder is what's 'left over' in division, which isn't wrong, but I also think it sometimes easiest to think of it as 'how many more than the nearest (lower) multiple of the divisor?'

So 31 has remainder 3 when divided by 7 because the nearest multiple of 7 (less than 31) is 28. 31 is 3 more than that, so it has a remainder of 3.

31 = 4(7) + 3