document.write( "Question 909060: The product of four prime numbers (they don't necessarily need to be all different) is the same as ten times the sum of these prime numbers.
\n" ); document.write( "What are all the possibilities for those four prime numbers?\r
\n" ); document.write( "\n" ); document.write( "The only possiblity that I've found so far is this:
\n" ); document.write( "2+3+5+5 = 15 and
\n" ); document.write( "2*3*5*5 = 150\r
\n" ); document.write( "\n" ); document.write( "What are other possibilities? How can I find out? Is there a formula?
\n" ); document.write( "In case, there are no more possibilities, what is the proof?\r
\n" ); document.write( "\n" ); document.write( "Thank you very much, all help is greatly appreciated!
\n" ); document.write( "P.S. Sorry for my English, I hope the question is understandable. \n" ); document.write( "

Algebra.Com's Answer #551550 by rothauserc(4718) $\"\"$ $\"About$

You can put this solution on YOUR website!
look at this as a prime factorization problem
\n" ); document.write( "let x be an integer with four prime factors, then we know that
\n" ); document.write( "x = a^p * b^q * c^r * d^s where a,b,c,d are prime factors and p,q,r,s are their powers, then
\n" ); document.write( "The number of factors of x will be expressed by the formula (p+1)(q+1(r+1)(s+1). NOTE: this will include 1 and x itself.
\n" ); document.write( "We are told that x=abcd, where a, b, c, and d are four distinct prime numbers. According to the above the # of factors of x including 1 and x is (1+1)(1+1)(1+1)(1+1)=16. Excluding 1 and x the # of factors is 16-2=14.
\n" ); document.write( "This exercise assumes that each prime factor is distinct, this gives you an upper bound on the number of prime factors of x.\r
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