Many whole numbers can be broken up into a run of two or more consecutive whole numbers — for instance 15 = 4 + 5 + 6. Which of these numbers cannot be broken up in that way?

A run of $k$ consecutive numbers starting at a adds to $k\times (2a+k-1)÷2$. Of the two factors $k$ and $2a+k-1$, one is always odd, and once $k$ ≥ 2 that odd factor is bigger than 1. So any number written this way must have an odd factor bigger than 1 — which rules out the powers of 2. Here $64=2^6$, so no run works. The other three all do: 27 = 13 + 14, 45 = 22 + 23 and 100 = 18 + 19 + 20 + 21 + 22. Choosing 100 because it looks 'round' is the usual slip; roundness in base ten says nothing about odd factors.