Why powers of 2?

Let \(n\) be a positive integer randomly chosen between the interval \(\left [ 1, \text{lcm} \left (2,3,4,\ldots , 2^{100} \right) \right ] \).

Let \(P \) denote the probability that \(1 + 2 + \cdots + n\) is divisible by \(2^{10} \).

What is \( \dfrac 1P \)?


Inspiration.

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