Question 1205841
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Find the difference {{{ y - x }}}, where x and y are positive integers and {{{ 0 = 7x^3 - x^2y^2 + 14399 }}}.
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<pre>
Your starting equation is 

    {{{0 = 7x^3 - x^2*y^2 + 14399}}}.


Factor 14399 = {{{7*11^2*17}}}.  Re-write equation in this equivalent form

    {{{x^2*y^2 - 7x^3}}} = {{{7*11^2*17}}}.


Right side is divisible by 7, and one term in the left side is a multiple of 7;
hence, the term {{{x^2*y^2}}} is divisible by 7.


So, I write y = 7z, where z is some positive integer number.

    I can not write x = 7z, since then the degree of 7 will be too great on the left side.


Then the last equation takes the form

    {{{x^2*(7z)^2 - 7x^3}}} = {{{7*11^2*17}}},  or

    {{{x^2*7^2*z^2 - 7x^3}}} = {{{7*11^2*17}}}.


Cancel factor 7 in both sides and get

    {{{7x^2*z^2 - x^3}}} = {{{11^2*17}}},

    {{{x^2*(7z^2-x)}}} = {{{11^2*17}}}.


Recall about the uniqueness of decomposition of integer numbers into the product of primes.

It implies  that  x= 11.


Then  7z^2 - x = 17;  7z^2 - 11 = 17;  7z^2 = 17 + 11 = 28;  z^2 = 28/7 = 4;  z= {{{sqrt(4)}}} = 2.


Thus x= 11;  z= 2.  Hence, y = 7z = 7*2 = 14.


The difference y-x is  14-11 = 3.


<U>ANSWER</U>.  y-x = 3.
</pre>

Solved.


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I understand that I am helping somebody to solve a problem of a Math Olympiad level.