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Is there a quick way to prime factorize 50!.

I wrote down all the numbers and then factorized, but that takes way too long.

Chirag Maheshwari
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  • I suppose that depends on what you consider "quick" but for each prime less than $50$ just look at the factors of it and how many times the prime appears in each factor individually and add. For instance, $50!$ has $5$ appearing as a factor a total of $12$ times (once each for $5,10,15,20,25,\dots,45,50$ and an additional time for $25,50$*). You can repeat this process for each prime less than $50$. – JMoravitz Sep 28 '19 at 15:14

2 Answers2

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I would list the primes less than $50$ and count how many times each one appears in $50!$

For example $$47,43,41,37,31,29$$ appears only once. $$23,19,17$$ appears two times each. $$13$$ appears three times. $$11$$ appears four times. $$7$$ appears $8$ times. (49 counts twice ) and so forth.

Mohammad Riazi-Kermani
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There is a quick way to do this. The number of times a prime number $n$ divides $k!$ is given by- $$N= {\sum_{i=1}^{\infty}}\left\lfloor{k\over n^i}\right\rfloor$$ For example, the power of $2$ in $20!$ will be $\left\lfloor{20\over 2}\right\rfloor+\left\lfloor{20\over 4}\right\rfloor+\left\lfloor{20\over 8}\right\rfloor+\left\lfloor{20\over 16}\right\rfloor$=10+5+2+1=18.

Hence $2^{18}$ divides $25!$

You would have to do this for all prime numbers till $k!$ though.

NOTE: The formula doesn't work if n isn't prime

Sam
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