SOLUTION: Find the product of x•y, where x and y are positive integers and {{{ 0=3x^4-x^3y-9317 }}}

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Question 1189711: Find the product of x•y, where x and y are positive integers and +0=3x%5E4-x%5E3y-9317+
Answer by ikleyn(52781) About Me  (Show Source):
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Find the product of x•y, where x and y are positive integers and 0+=+3x%5E4+-+x%5E3%2Ay-9317.
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Given equation is equivalent to

    3x^4 - x^3*y = 9317,  

or

    x^3*(3x - y) = 9317.


Integer number 9317 has the primary decomposition  9317 = 7*11^3.


From uniqueness of prime decomposition for integer numbers, we conclude that  

    EITHER  x = 1,  3x-y = 9317  OR  x = 11 and  3*11 - y = 7;  y = 33-7 = 26.


First option produces negative value of y; therefore, we discard it.

The second option gives the ANSWER to the problem  x= 11, y = 26.


The requested product x*y = 11*26 = 286.    ANSWER

Solved.