9TH DIGIT PALINDROMIC
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Find the smallest palindromic number of 9 digits whose root is also a palindromic.
- kfoozminus
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Re: 9TH DIGIT PALINDROMIC
$10001^2=100020001$
jannatul ferdows jenny
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- nafistiham
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Re: 9TH DIGIT PALINDROMIC
Probably, that is correct in this way,kfoozminus wrote:$10001^2=100020001$
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
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- Fahim Shahriar
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Re: 9TH DIGIT PALINDROMIC
Though it's already done with nafis tiham's observation, I'd like to give a solution.
Suppose the root of the palindrome is $10000a+1000b+100c+10b+a = 10001a+101b+100c$
Square it, and take (mod 4). We will get
$a^2+b^2+ab=x^2 (mod 4)$
[We know $x^2=0 or 1 (mod 4)$]
$a$'s least value is 1(it's leftmost digit) and $b$ is either 0 or an even number and its least value is 0. For this $c \leq 2$ and least value 0.
So the root is 10001. And the number is 100020001.
Suppose the root of the palindrome is $10000a+1000b+100c+10b+a = 10001a+101b+100c$
Square it, and take (mod 4). We will get
$a^2+b^2+ab=x^2 (mod 4)$
[We know $x^2=0 or 1 (mod 4)$]
$a$'s least value is 1(it's leftmost digit) and $b$ is either 0 or an even number and its least value is 0. For this $c \leq 2$ and least value 0.
So the root is 10001. And the number is 100020001.
Name: Fahim Shahriar Shakkhor
Notre Dame College
Notre Dame College
- kfoozminus
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Re: 9TH DIGIT PALINDROMIC
$9999^2$ is a $8$ digit number, $10000^2$ isn't palindromic, so...nafistiham wrote:[Actually, I wondered, how it is the smallest. So, shared my observation.]
jannatul ferdows jenny
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