September 22, 2017

Horse 2324 - x²-1 Is Divisible By 8 For All Odd x


I was speaking to someone earlier this week and I happened to make the observation on the fly that:

If you take any odd number and square it, then take away 1, then that result will always be divisible by 8.

On the face of it, that seems like some absolutely mind blowing thing; but in reality I just happened to notice something really really quickly because I live in a world where playing with arithmetic is commonplace.
Cast your minds back to high school algebra and consider the rather short expression...


If you'll remember from algebra, the full binomial expansion of this is...


This is easy enough to prove...

x² + 1x - 1x - 1 (First, Inside, Outside, Last)**
x² - 1

The thing is though that for every odd number x, the two numbers either side, that is x-1 and x+1 are both even. That sounds so elementary that even John Watson in Kindergarten who is going to elementary school should realise how elementary it is.
The next leap of logic though is that one of those numbers, either x-1 or x+1 is a multiple of four, because every second even number is a multiple of four.

Without even telling you what say, 105,827 x 105,827 is, I can tell you that it equals whatever (105,826 x 105,828) +1 is; I already know that because the last two digits of 105,828 are divisible by four, then 105,828 is and because the other number is even, whatever the answer is, is going to be divisible by 8.

For the record, 105,827 x 105,827 = 11,199,353,929 and...
11,199,353,929 - 1 = 11,199,353,928. (which is divisible by 8).
11,199,353,928 = 105,826 x 105,828

Sounds stupid?

If you test the smallest odd number (because x²-1 has a lower limit), then you get when x = 1

1² + 1 - 1 - 1
= 0

0 is is divisible by 8.


Incidentally, that general suggestion that the square of a number minus 1 equals the two numbers either side multiplied together, also holds true for 0.

= -1

-1 x +1 = -1

That's probably a trivial case though.

*I don't know what factorial "MATHS AHEAD" would be.
**Also called the "Foil Method" for ease of remembering.

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