Puzzles
1 December
The geometric mean of a set of \(n\) numbers can be computed by multiplying together all the numbers then computing the \(n\)th root of the result.
The factors of 4 are 1, 2 and 4. The geometric mean of these is 2.
The factors of 6 are 1, 2, 3, and 6. The geometric mean of these is \(\sqrt{6}\).
The geometric mean of all the factors of today's number is 22.
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The geometric mean of a number \(N\) is always \(\sqrt{N}\). You can see why this is true by considering the pairs of factors that multiply to make the number:
if \(N\) if not square and has \(k\) pairs of factors, then the product of these factors is \(N^k\), so the geometric mean is \((N^k)^{1/(2k)}=N^{1/2}\);
if \(N\) is square and has \(k\) pairs of factors plus the square root, then the product of its factors is \(N^k\sqrt{N}=N^{k+1/2}\), so the geometric mean is \((N^{k+1/2})^{1/(2k+1)}=N^{1/2}\).
Interestingly, this means that the geometric mean of the factors of a number is an integer only when the number is square.
Therefore the only number whose factors have a geometric mean of 22 is \(22^2\), or 484.
