Legendre's Formula states that
Find the largest integer for which divides
Using the first form of Legendre's Formula, substituting and gives which means that the largest integer for which divides is .
Using the second form of Legendre's Formula, substituting and gives The number when expressed in Base-2 is . This gives us . Therefore, which means that the largest integer for which divides is .
We use a counting argument.
We could say that is equal to the number of multiples of less than , or . But the multiples of are only counted once, when they should be counted twice. So we need to add on. But this only counts the multiples of twice, when we need to count them thrice. Therefore we must add a on. We continue like this to get . This makes sense, because the terms of this series tend to 0.
Let the base representation of be where the are digits in base Then, the base representation of is Note that the infinite sum of these numbers (which is ) is
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