in base 10 a number is even iff its ones digit is an even number. Show this statement is true when 10 is replaced by any even base.
A 2 digit number n with tens digit t and units digit u is equal in value to 10*t+u. Since 10*t is even irregardless of what you choose for t, the evenness or oddness of n depends only on u.
3 digit numbers will be 100*h+10*t+u, and by the same argument, its parity depends only on u.
This holds for any number of digits in base 10.
In base r, a 2 digit number is r*d1+d0, with d1 being the 'tens' digit and d0 being the units digit. If r is even, the parity depends only on d0.
This argument extends to any number of digits, and any base r that's even.