<PRE><B>Question 1123295</B>
&lt;br&gt;
40 = (2^3)(5^1)&lt;br&gt;
For each power of 40, you need 3 prime factors of 2 and 1 prime factor of 5.&lt;br&gt;
In 80!, the number of factors of 5 is 16+3 = 19.&lt;br&gt;
In 80!, the number of factors of 2 is 40+20+10+5+2+1 = 78.&lt;br&gt;
The number of prime factors of 2 in 80! is more than 3 times the number of prime factors of 5; that means the number of prime factors of 5 is what limits the power of 40 that divided into 80! gives a whole number answer.  So&lt;br&gt;
Answer: 19&lt;br&gt;
I verified that answer using the free online PARI calculator.&lt;br&gt;
80!/40^19 yielded a whole number result:&lt;br&gt;
26036816796936837743418994695525373450649442872369309980825872138021455341178861578616832&lt;br&gt;
80!/40^20 yielded a result that is NOT a whole number:&lt;br&gt;
3254602099617104717927374336940671681331180359046163747603234017252681917647357697327104/5&lt;br&gt;
Note that this latter result indeed shows us that we are one factor of 5 short of getting a whole number answer.
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