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Power of three in a factorial

MediumCombinatorics~5m

Write 100!=3k⋅m100! = 3^{k}\cdot m where mm is not divisible by 33. What is the exponent kk (the highest power of 33 dividing 100!100!)?

Show hints (2)+
  1. Legendre's formula sums ⌊n/pi⌋\lfloor n/p^i\rfloor over ii.
  2. Add ⌊100/3⌋+⌊100/9⌋+…\lfloor 100/3\rfloor + \lfloor 100/9\rfloor + \dots

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