Divisors and the Geometric Mean

Solution

Let the divisors of n be d₁, d₂, ..., d_k. The geometric mean of the product of the divisors is (d₁d₂...d_k)^1/k.

If n is not a square number, then the divisors can be organized into pairs whose product is n (e.g. the divisors of 12 are 1 and 12, 2 and 6, 3 and 4). In this case, k is an even number, so let k = 2m. Ordering the divisors into pairs, we can write the geometric mean as:

(d₁d₂)^1/k(d₃d₄)^1/k...(d_m-1d_m)^1/k
= n^1/kn^1/k...n^1/k
= n^m/k
= n^1/2.

If n is a square, then it its square root as one divisor, and all the other divisors come in pairs as before. Write k = 2m + 1, and the geometric mean is:

(d₁d₂)^1/k(d₃d₄)^1/k...(d_m-1d_m)^1/kn^1/2k
= n^1/kn^1/k...n^1/kn^1/2k
= n^m/kn^1/2k
= n^(2m+1)/2k
= n^1/2

In all cases, then, the geometric mean of the divisors of n is equals to the square root of n.