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Number Theory Level pending

Find the highest power of 3 which divides 1000 ! 1000! .

495 495 498 498 492 492 499 499

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1 solution

Akshat Sharda
Feb 6, 2016

= 1000 3 + 1000 3 2 + 1000 3 3 + 1000 3 4 + 1000 3 5 + 1000 3 6 = 333 + 111 + 37 + 12 + 4 + 1 = 498 \begin{aligned} & = \left \lfloor \frac{1000}{3} \right \rfloor + \left \lfloor \frac{1000}{3^2} \right \rfloor + \left \lfloor \frac{1000}{3^3} \right \rfloor + \left \lfloor \frac{1000}{3^4} \right \rfloor + \left \lfloor \frac{1000}{3^5} \right \rfloor + \left \lfloor \frac{1000}{3^6} \right \rfloor \\ & = 333+111+37+12+4+1 \\ & = \boxed{498}\end{aligned}

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