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What is the greatest positive integer n such that \(3^n\) is a factor of \(200! \) ?

 

Help appreciated!

 Oct 13, 2020
 #1
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200!= 2^197 * 3^97 * 5^49 * 7^32 * 11^19 * 13^16 * 17^11 * 19^10 * 23^8 * 29^6 * 31^6 * 37^5 * 41^4 * 43^4 * 47^4 * 53^3 * 59^3 * 61^3 * 67^2 * 71^2 * 73^2 * 79^2 * 83^2 * 89^2 * 97^2 * 101 * 103 * 107 * 109 * 113 * 127 * 131 * 137 * 139 * 149 * 151 * 157 * 163 * 167 * 173 * 179 * 181 * 191 * 193 * 197 * 199

 

The greatest positive n = 97 or 3^97.

 
 Oct 13, 2020
 #2
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You can also do it this way:

 

n = [200/3 + 200/3^2 + 200/3^3 + 200/3^4] = 66 + 22  + 7 + 2 =97 [You take the integer part only].

 
 Oct 13, 2020

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