Question 1150079
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<pre>
The prime decomposition of the number 1215000 is


    1215000 = {{{3^5}}}.{{{5^4}}}.{{{2^3}}}.


Comparing it with  {{{a*b^4*c^3}}}, you see that


    either  a= 3^5,  b= 5,    c= 2,      with  a+b+c = 3^5 + 5 + 2   = 250,

    or      a= 3^2,  b= 5,    c = 2*3,   with  a+b+c = 3^2 + 5 + 2*3 =  20,

    or      a = 3,   b= 5*3,  c = 2,     with  a+b+c = 3   + 5*3 + 2 =  20.


So, a+b+c is maximal and equal to 250  at  a= 3^5,  b= 5  and c= 2.    <U>ANSWER</U>
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Solved.