## Ones

cinelli
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### Ones

What is the smallest positive whole number with the property that when the digit 1 is appended to both ends, the new number is 99 times the original?

Cinelli

GoSeigen
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### Re: Ones

cinelli wrote:What is the smallest positive whole number with the property that when the digit 1 is appended to both ends, the new number is 99 times the original?

Cinelli

Spoiler:

Let our number be x. Then:

`10x+10^n+1=99x10^n+1=89x`

where n is a whole number.

So some number starting with a 1, ending with a 1, and with just enough zeros to become exactly divisible by 89 is the one we want. An easy way to find the answer is to do a long division: start with 101 but add extra zeros repeatedly between the ones until 89 divides exactly. I won't do the working here, but the answer quickly drops out as:

`x=112359550561797752809`

GS

GoSeigen
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### Re: Ones

GoSeigen wrote: just enough zeros to become exactly divisible by 89 is the one we want.

Just rereading this I note it's not really precise. What I meant was: "just enough zeros to become exactly divisible by 89 is the one we want for the left hand side of the equation."

GS

cinelli
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### Re: Ones

Quite right, GoSeigen, with a very concise explanation.

Cinelli