Question

Jillian loves math tricks. She asked her friend to choose any prime number greater than 3, square it, then subtract 4,then divide the new result by 12 and record the remainder. She then told her friend that the remainder was 9. How could she be sure that the remainder was 9 without know which prime number her friend had picked?

Answer 1

The reason it works is this:

Let n be a positive odd number greater than or equal to 3. There is a positive integer i so that n = (2i+3). Then n2 - 4 = 4i2 + 12i + 5 = 4i(i + 3) + 5

Now if i is divisible by 3, then there exists some other integer a such that 3a = i, so
n2 - 4 = 12a(3a + 3) + 5. So if we divide by 12, we get 5 as a remainder.

If (i - 1) is divisible by 3, then there exists some integer b such that 3b = i - 1 or in
other words, i = 3b + 1. So
n2 - 4
= 4(3b + 1)(3b + 4) + 5
= 4(9b2 + 15b + 4) + 5
= 12(3b2 + 5b + 1) + 9.

So here we get 9 as a remainder if we divide by 12.

The last possibility is that (i - 2) is divisible by 3, so there exists some integer c
such that 3c = i - 2, or i = 3c + 2. So
n2 - 4
= 4(3c + 2)(3c + 5) + 5
= 4(9c2 + 21c + 10) + 5
= 12(3c2 + 7c + 3) + 9

So here again we get 9 as a remainder.

Done..

Answer 2

Steps From: http://mathcentral.uregina.ca/QQ/database/QQ.09.06/eliseo1.html