A new December and a new bunch of puzzles from mscroggs.co.uk.

I think the example is also a hint.

A number with 1 factor would be 1. And that’s not 13.

A number with 2 factors would be a prime. Number p, factors 1 and p. The geometric mean would be (1*p)^1/2. We can’t get 13 to fit there.

A number with 3 factors has a special property. There’s a square here somewhere. The number, p², has factors 1, p and p². This accounts for p² = 1 * p² = p * p, the 2 different ways to write p². Like, 9 = 1*9 = 3*3. The geometric mean works out to be p, isn’t that neat? So my guess is that we’re looking for 13² = 169.