Question 1114882
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If m and k are integers, what are all the solutions to the equation {{{9(7^k+7^(k+2))= 5^(m+3)+5^m}}}
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{{{9*(7^k+7^(k+2))}}} = {{{5^(m+3)+5^m}}}  ====>


{{{9*7^k*(1 + 7^2)}}} = {{{5^m*(5^3 + 1)}}}


{{{9*7^k*50}}} = {{{5^m*126}}}  ====>   Divide both sides by 9  ====>


{{{7^k*50}}} = {{{5^m*14}}}  ====>


{{{50*7^k}}} = {{{2*7*5^m}}}  ====>  Comparing prime factors and their degrees and using the theorem on uniqueness 
                         of decomposition integers into the product of their prime factors


<U>Answer</U>.  m = 2,  k = 1.
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