How do you prove that a number is even?
Proving that a number is even can be easy for a 4th grader, hard for a smart high-schooler, worthy of a math Ph.D. at Harvard, or beyond the current abilities of all the world’s mathematicians put together.
“A number” may be given to us in many different ways. It can be the number of objects defined by some combinatorial criterion, it can be explicitly given by some simple or complicated formula, it can be the number of solutions of an equation, and many other variations on those themes.
So, it's impossible to answer such questions at this wild level of generality. “Looking at the last digit” is a fine idea if the number is “24”, but not so much if it's “the number of winning opening moves in the game of Go”.
Proving that a number is even can be done, for example, by finding an explicit pairing between the objects counted by the number. The pairing is often a reflection of some underlying symmetry, sometimes an obvious one, sometimes a devilishly tricky one.
In other instances, a number may be proven even by writing it as a product and confirming that one of the factors is itself even. Or writing it as a sum and showing that the number of odd terms is itself even. Or by reducing everything mod 2 and working in the field of two elements.
Some examples:
- Show that the number of pawns in chess is even.
- At a party, everyone shakes hands with everyone else. Show that the number of handshakes is even when the number of people is one of 4, 5, 8, 9, 12, 13, 16, 17 and so on (what do I mean by “and so on”?)
- A drunkard starts at a lamp post, walks 100 steps each of which is to the right or to the left, and ends up back at the lamp post. Show that the number of possible ways this can happen is even.
- Given two natural numbers n, k, find a simple way to know if the binomial coefficient \binom{n}{k} is even or odd.
- Show that the number of elements in a non-abelian finite simple group is even.
- Determine if the number of non-trivial roots of the Riemann zeta function, to the left of the critical line and in the upper half plane, is even (assuming it is finite).