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All odd digit (Posted on 2020-10-05) Difficulty: 2 of 5
Find all positive integers n such that the decimal representation of n2 consists of odd digits only.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution a proof | Comment 2 of 6 |
I'm sure there is a more elegant way to prove this, but ...

Assertion:  for all odd n > 3, the base 10 representation of n^2 has an even digit in the ten's column.
(In fact, even 1 and 9 contain even digits if you count leading zeros, which we don't).
Proof
Suppose not.
Suppose that for some odd n>3, n^2 has an even digit in the ten's column, but (n+2)^2 contains only odd digits.
(n+2)^2 = n^2 + 4n + 4

Consider the pattern (for odd n) of the final digit of n, and the final two digits of n^2, and (4n+4). 
The ten's digit will be represented by 'e' for even, or the letter 'o' for odd.
n  n^2  (4n+4)  sum (n^2+4n+4)
1    e1    e8      e9
3    e9    o6      e5
5    e5    e4      e9
7    e9    o2      e1
9    e1    e0      e1
  Posted by Larry on 2020-10-05 09:58:51
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