The Greatest

Number Theory Level 2

Find the greatest positive integer x x such that 2 3 6 + x 23^{6+x} divides 2000 ! 2000!


The answer is 83.

Danish Ahmed by Danish Ahmed , 24, India

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1 solution

Pranjal Jain
Feb 22, 2015

Since 23 is a prime number,

Greatest power of 23 23 in 2000 ! 2000! will be ⌊ 2000 23 ⌋ + ⌊ 2000 2 3 2 ⌋ + ⋯ \left\lfloor\dfrac{2000}{23}\right\rfloor +\left\lfloor\dfrac{2000}{23^2}\right\rfloor +\cdots = 86 + 3 + 0 + ⋯ = 89 =86+3+0+\cdots=89

x + 6 = 89 ⇒ x = 83 x+6=89\Rightarrow x=83

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