# 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 $$\dfrac 1P$$?

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