Thursday 26 January 2012

Any Power of 10 Minus One is Divisible by 9

(10^x - 1) / 9 = y, x member of natural numbers implies y is a natural number.

Proof.
When x = 1 then y = (10^1 - 1) / 9 = (10 - 1) /  9 = 9 / 9  = 1
Assume that (10^a -1) / 9 = b for some a, b members of the natural numbers.
Now consider
c = (10^(a+1) - 1) / 9
c = (10^a.10 - 10 + 9) / 9
c = (10.(10^a - 1) + 9 ) / 9
c =10.(10^a - 1)/9 + 9/9
c = 10.b + 1
Since b is a natural number then so is c.
So, by induction, we have shown that (10^x  - 1) / 9 is a natural number or, equivalently, 10^x - 1 is always divisible by 9.

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