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\n" ); document.write( "40 = (2^3)(5^1)
\n" ); document.write( "For each power of 40, you need 3 prime factors of 2 and 1 prime factor of 5.
\n" ); document.write( "In 80!, the number of factors of 5 is 16+3 = 19.
\n" ); document.write( "In 80!, the number of factors of 2 is 40+20+10+5+2+1 = 78.
\n" ); document.write( "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
\n" ); document.write( "Answer: 19
\n" ); document.write( "I verified that answer using the free online PARI calculator.
\n" ); document.write( "80!/40^19 yielded a whole number result:
\n" ); document.write( "26036816796936837743418994695525373450649442872369309980825872138021455341178861578616832
\n" ); document.write( "80!/40^20 yielded a result that is NOT a whole number:
\n" ); document.write( "3254602099617104717927374336940671681331180359046163747603234017252681917647357697327104/5
\n" ); document.write( "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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