# Square remainders

NN2rem NN2rem
1117491
2448644
3999819
4164101004
5251111211
6360121440

### "All squares, on division by twelve, leave a remainder which is also a square."

rem = remainder mod twelve

The statement is also true, by the way, if we replace the word "twelve" by "five", "sixteen", or "twenty-four".

Can you find other solutions?

# Quick Squares

(NB "10" means any number base)

### A handy rule for squaring numbers

(mental, or pencil & paper)

1. Find the difference, "d" between the number, "n", and the next multiple of "10" (above or below n)
2. If you needed to add the difference, d, to get the multiple, also form the number n-d;
if you needed to subtract the difference, also form the number n+d.
3. Multiply these numbers (n+d) and (n-d) and then add the square of the difference, d2, to this answer.

Which says that: n2 = (n-d)(n+d) + d2

Here are some examples in numbers:

1. (base ten)
n = "38"; next multiple of 10 is 38+2, i.e. 40; this is n+d, so d = 2
also form n-d: 38 - 2 = 36.
Square of 38 is 36 x 40 + 22 = 1440+4 = 1444
2. (base seven)
n = "36" ; next multiple is 40, or 36 + 1
also form 36 - 1 = 35.
Square of 36 is 35 x 40 + 12 = 2060 + 1 = 2061.
3. (any base greater than 4)
n = "12" ; nearest multiple is 10, or 12 - 2;
also form 12 + 2 = 14.
Square of 12 is 14 x 10 + 22 = 140 + 4 = 144

And the proof? [ (n-d)(n+d) ] +d2 =[ n2 -nd +nd + d2 ] + d2

= [n2 - d2] + d2 = n2.