Odd or Even in Different Bases
Can 27 represent an even number in some scale? Can 37? Can 72 represent an odd number
in some scale? Can 82?
We want to know if for same base b, (27)b can be written as an even number. We
will try both odd and even bases. Let b=2n+1 represent odd bases and b=2n
represent even bases.
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27b  22n  1  71
27b  22n   71
 4n  2  7
 4n  7
 4n  9
 4n  7
 4n  8  1
 4n  6  1
 22n  4   1
 22n  3  1
Let 2n  4  l.
Let 2n  3  l.
27b 2n 1  2l  1
27b 2n  2l  1
For both even and odd bases,
27 will still be odd.
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37b  32n  1  71
37b  32n   71
 6n  3  7
 6n  7
 4n  10
 6n  6  1
 22n  5
 23n  3  1
Let 2n  5  l.
37b 2n 1  2l
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37 can represent an even number
for any odd base. For any even base, 37 will
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remain odd.
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72b  72n  1  21
 14n  7  2
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Let 3n  3  l.
37b 2n  2l  1
72b  72n  21
 14n  9
 14n  2
 14n  8  1
 27n  1
 27n  4   1
Let 7n  4  l.
Let 7n 1 l.
72b 2n 1  2l  1
72b 2n  2l
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72 can represent an odd number for any even base. For any odd base, 72 will remain
even.
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82b  82n  1  21
 16n  8  2
 16n  2
 16n  10
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82b  82n  21
 28n  1
 28n  5
Let 8n  5  l.
Let 8n 1 l.
82b 2n 1  2l
82b 2n  2l
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For both even and odd bases, 82 will still be even.
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