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Labor of Division (Episode 1)

So dividing by 13 is the same as multiplying by 1321528399 and then dividing by 2^34. That means that 2^34 divided by 13 would be 1321528399, right? In fact, it’s 1321528398.769 and change. Pretty close, but we’re not optimizing horseshoes, so how can we be sure this works all the time?