What is the Relationship Between Even Bases and Even Numbers in a Number System?

[Noxide]

Homework Statement

Whenever base b is even (b=2k for some integer k) a number H=(d_(n-1)d_(n-2)...d₁d₀)_b is even <=> d₀ is even.

Homework Equations

The Attempt at a Solution

I have formulated a proof for the forward direction (=>) but I am having trouble getting started on a proof for the backwards direction (<=) starting with the fact that d₀ is even.
Any advice would be much appreciated. I know all the definitions I'm just unsure how to build from d₀ is even.

 

[Mark44]

Homework Statement

Whenever base b is even (b=2k for some integer k) a number H=(d_(n-1)d_(n-2)...d₁d₀)_b is even <=> d₀ is even.

Homework Equations

The Attempt at a Solution

I have formulated a proof for the forward direction (=>) but I am having trouble getting started on a proof for the backwards direction (<=) starting with the fact that d₀ is even.
Any advice would be much appreciated. I know all the definitions I'm just unsure how to build from d₀ is even.

You don't show what you've done, so this might or might not be useful.

H=(d_(n-1)d_(n-2)...d₁d₀)_b
= d_(n-1) * b^{n - 1} + d_(n-2) * b^{n - 2} + ... + d₁*b + d₀

You have d₀ being even. What can you say about b raised to any positive power? Is it even or odd? What about multiples of even or odd numbers?

 

[Noxide]

even * even = even always
odd * even = even always
even + odd = odd

I think my problem was identifying the problem correctly

I rewrote it this way

b = 2k, k is a natural number, H=(d(n-1)d(n-2)...d1d0)b is even <=> do is even

shouldnt be too hard from there