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Ones

cinelli
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Ones

#120738

Postby cinelli » February 27th, 2018, 12:36 pm

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

#120916

Postby GoSeigen » February 28th, 2018, 12:06 am

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=99x
10^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

#120991

Postby GoSeigen » February 28th, 2018, 10:29 am

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

#121367

Postby cinelli » March 1st, 2018, 2:47 pm

Quite right, GoSeigen, with a very concise explanation.

Cinelli


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