Is $N_{10}$ even or odd?

[Ackbach]

Here is this week's POTW:

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Let $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots +1/a_n=1$. Determine whether $N_{10}$ is even or odd.

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[Ackbach]

Re: Problem Of The Week # 230 - Aug 25, 2016

This was Problem A-5 in the 1997 William Lowell Putnam Mathematical Competition.

No one answered this week's POTW. The solution, attributed to Kiran Kedlaya and his associates, follows:

Spoiler

We may discard any solutions for which $a_1 \neq a_2$, since those come in pairs; so assume $a_1 = a_2$. Similarly, we may assume that $a_3 = a_4$, $a_5 = a_6$, $a_7 = a_8$, $a_9=a_{10}$. Thus we get the equation
\[
2/a_1 + 2/a_3 + 2/a_5 + 2/a_7 + 2/a_9 = 1.
\]
Again, we may assume $a_1 = a_3$ and $a_5 = a_7$, so we get $4/a_1 + 4/a_5 + 2/a_9 = 1$; and $a_1 = a_5$, so $8/a_1 + 2/a_9 = 1$. This implies that $(a_1-8)(a_9-2) = 16$, which by counting has 5 solutions. Thus $N_{10}$ is odd.