This may be OT since it doesn't really have anything to do with Perl,
but one of your original assumptions is wrong.

In an even base a integer ending in an even digit must be even because
the previous digits are multiplied by a power of an even base .
Leaving the last digit to decide the parity.

In an odd base the total of the digits must be even for the integer to
be even because all of the digits are multiplied by a power of an odd base so
that the that the parity of the integer depends on parity of all the digits.

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