Melody

I didn't understand the question either.

ibhar:

Where is your response to Textot?     

Answerers want interaction!

Thanks Textot   

Alan proved what you asked and you did not even bother to thank him.

You did not even give him a point.

Why would any serious answerers bother with you?

Hi jsaddern and catmg,     

I will try and explain how I was thinking.

I know this.

\((10^{n+1}+1)>(a_n \cdot 10^{n} + a_{n-1} \cdot 10^{n-1} + \ldots + a_0 \cdot 10^0)\)

So I know that if    \((10^{n+1}+1)(a_n \cdot 10^{n} + a_{n-1} \cdot 10^{n-1} + \ldots + a_0 \cdot 10^0)\)   is a perfect sqare then

\((10^{n+1}+1)\)     must have at least one perfect square factor.   Can you see that?

So let         \((10^{n+1}+1) =a^2b\)         where a and b are rational positve numbers

then   \((a_n \cdot 10^{n} + a_{n-1} \cdot 10^{n-1} + \ldots + a_0 \cdot 10^0)\)    must be     bc^2   where b and c are rational positive numbers

and  a>c

That would make    \((10^{n+1}+1)(a_n \cdot 10^{n} + a_{n-1} \cdot 10^{n-1} + \ldots + a_0 \cdot 10^0)= a^2b*bc^2=(abc)^2\)

So I googled online for a number of the form  10^n+1  with a perfect square factor.

I found this site:

I think I can disprove it with an example.   n=10    (Edit correction, I had previously said n=11)

10^11+1 = 121*826446281

so

\(\sqrt{(10^{11}+1)*(826446281)} = \sqrt{121^2*826446281^2}=121*826446281 \)

\((10^{n+1}+1)(a_n \cdot 10^{n} + a_{n-1} \cdot 10^{n-1} + \ldots + a_0 \cdot 10^0)\\ when\;\; n=10\\ (10^{11}+1)(826446281)\qquad \text{is a perfect square}\)

sqrt((10^11+1)*826446281 = 9090909091

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Again I reread your question, it says 'between 0 and 9'  Is that supposed to include 0 and 9?

My answer uses 0.

I suppose I wasn't meant to but is 9 also really not allowed?