Pairs to make fours

Take a pair of even numbers and add them. Is the sum divisible by 4? Sometimes: 12 + 16 = 28, yes; 4 + 6 = 10, no. When does it work?

The key is to split even numbers into two kinds, by what they leave when divided by 4:

Every even number is one or the other.

So there are three cases to check.

Case 1: both are multiples of 4. Write them 4p and 4q. Then 4p + 4q = 4(p + q): always a multiple of 4. Picture p rows of four beside q rows of four: together, (p + q) full rows. Examples: 12 + 16 = 4(3 + 4) = 28; 16 + 28 = 4(4 + 7) = 44.

Case 2: neither is a multiple of 4. Write them 4p + 2 and 4q + 2. Then (4p + 2) + (4q + 2) = 4p + 4q + 4 = 4(p + q + 1): also always a multiple of 4. The two spare 2s join up to make one more full row of four. Examples: 2 + 6 = 8; 6 + 10 = 16; 22 + 6 = 28.

Case 3: one of each. 4p + (4q + 2) = 4(p + q) + 2. There's a leftover 2 with nothing to pair with, so the sum is never a multiple of 4. Examples: 4 + 6 = 10; 8 + 10 = 18.

Look at the shape of that answer: two of the same kind work; one of each doesn't. That's exactly the pattern of odd and even, one level up.

And notice the method. Algebra proved all three cases in three lines, and the row pictures showed the same argument in a way you can see. Both are proofs; they just speak differently.