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
As you can see, n = 3^97
+26367 What is the greatest positive integer n such that 3 to the power of n is a factor of 200!
\(\begin{array}{|rcll|} \hline n &=& \lfloor \dfrac{200}{3}\rfloor + \lfloor \dfrac{200}{3^2}\rfloor+\lfloor \dfrac{200}{3^3}\rfloor+\lfloor \dfrac{200}{3^4}\rfloor \\\\ n &=& 66+ 22+7+2 \\ \mathbf{n} &=& \mathbf{97} \\ \hline \end{array}\)
+118608 What is the greatest positive integer n such that 3 to the power of n is a factor of 200!. (that exclamation mark is a factorial by the way)
Here is a neat little way to do it. See if you can work out why it works.
3,9,27,81 are all powers of 3
How many multiples af 81 are there 2
How many multiples af 27 are there 7
How many multiples af 9 are there 22
How many multiples af 3 are there 66
2+7+22+66 = 97
