## Friday, January 26, 2007

### Fun Easy Number Theory Proof

I like to show this one to younger students who are disenchanted by algebra.

So, write down any 3 digit number (I started off with "so" because it seems like every math prof I ever had would start their sentences that way...)

Write those 3 digits down twice in the same order so that you have a 6 digit number.

For example:
123 ---> 123123
or
505 ---> 505505

This resulting number will ALWAYS be divisible by 7, 11 and 13 and will always result in the original 3 digit number.

Let's take 123123 to illustrate:

123123 divided by 7 = 17589
17589 divided by 11 = 1599
1599 divided by 13 = 123

So how do you prove this?

It's a surprisingly accessible proof.
The statement "divisible by 7, 11 and 13" is the same as saying "divisible by (7 times 11 times 13)"

"7 times 11 times 13" = 7 x 11 x 13 = 1001

So, really, we're saying the resulting 6 digit number will always be evenly divisible by 1001 (if you recall, evenly divisible means there are no remainders).

***
To state this in general terms:
Let a, b and c be integers between 0 and 9. We form a six digit number in the following way.

abcabc = a(10^5) + b(10^4) + c(10^3)+ a(10^2) + b(10^1)+ c(10^0)

This number will always be divisible by 1001 and the resulting answer will be equal to abc

***
Some reminders:
Any number to the power of zero is equal to 1.
Any number can be written as the sum of integers times powers of 10. For example:

123 = 1(10^2)+ 2(10^1)+ 3(10^0)
= 1(10^2)+ 2(10^1)+ 3(1)
= 1(100) + 2(10) + 3(1)
= 100 + 20 + 3
= 123

***

The proof only involves a lot of factoring:

abcabc = a(10^5) + b(10^4) + c(10^3) + a(10^2) + b(10^1) + c(10^0)
= a(10^5) + b(10^4) + c(10^3) + a(10^2) + b(10^1) + c
= a(10^5) + a(10^2) + b(10^4) + b(10^1) + c(10^3) + c
= a(10^2)[(10^3 +1)] +b(10^1)[(10^3 + 1)] +c[(10^3 + 1)]
= [a(10^2) + b(10^1) + c](10^3 +1)
= [a(10^2) + b(10^1) + c](1001)
= abc(1001)

In other words, abcabc = abc(1001) which means that abcabc is divisible by 1001 and dividing the number by 1001 leaves abc. Proof solved!

Labels: dasMobius said... cchang said...