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The question is the following: "Which is the highest power of 18 that divides 190! ?"

I seem to be under the impression that I don't know the "formula" correctly as this is my solution (which is so far wrong).

$18=3^22$ So I thought we'd look at the the 3's here as they are the highest prime factor of 18.

$\lfloor$180/3$\rfloor$+$\lfloor$180/$3^2$$\rfloor$+$\lfloor$180/$3^3$$\rfloor$+$\lfloor$180/$3^4$$\rfloor$=93

This is not the right answer but what is it that I do wrong?

Linus Jensen
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  • Do you want $190!$ or $180!$? – Angina Seng Aug 08 '17 at 18:26
  • You have to find the highest power of $3$ squared. You figured out that $3^{93}|180!$ and $3^{94}\not \mid 180!$ (assuming your calculations are correct). So $9^{46}$ is the highest power of $9$. Now if $2^{46}|180!$ (which it must as $2< 9$) we have the answer if $46$. – fleablood Aug 08 '17 at 19:18

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Your calculation gives you the number of $3'$s present in $180!$. But each $18$ requires two $3'$s, so the number of $18's$ you can have is $\left \lfloor \dfrac {93}{2} \right\rfloor=46$ .

Ovi
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