Distribution Of Divisor Sums Modulo 32

For each integer \(2 \le k \le n\), choose a divisor \(d_k\) of \(k\), uniformly at random from the set of divisors of k. We denote by \(P(n)\) the probability that \[d_2 + d_3 + \cdots + d_n\] is divisible by 32.

Amazingly, there exists a positive integer \(N\) such that for all \(n\ge N\), the value of \(P(n)\) is exactly \(\dfrac1{32}\). What is the smallest \(N\) for which this is true?

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