Replacing two numbers by their difference

Idea. Find a quantity that a move cannot change, at least modulo $2$.

Invariant. A move replaces the two numbers $a,b$ by $|a-b|$, so the sum $T$ of the numbers on the board changes by
$$-(a+b)+|a-b|=-2\min(a,b),$$
which is even. Hence the parity of $T$ never changes.

Conclusion. At the start $T=1+2+\dots+10=55$, which is odd. After nine moves, $T$ is the single remaining number, and it is odd.

Remark. The same argument works for $1,\dots,n$: the final number has the parity of $\frac{n(n+1)}2$.

Lesson. Invariant: find a quantity (often a parity) that every allowed move preserves.