SOLUTION: 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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Question 1114882: If m and k are integers, what are all the solutions to the equation 9%287%5Ek%2B7%5E%28k%2B2%29%29=+5%5E%28m%2B3%29%2B5%5Em
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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If m and k are integers, what are all the solutions to the equation 9%287%5Ek%2B7%5E%28k%2B2%29%29=+5%5E%28m%2B3%29%2B5%5Em
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9%2A%287%5Ek%2B7%5E%28k%2B2%29%29 = 5%5E%28m%2B3%29%2B5%5Em  ====>


9%2A7%5Ek%2A%281+%2B+7%5E2%29 = 5%5Em%2A%285%5E3+%2B+1%29


9%2A7%5Ek%2A50 = 5%5Em%2A126  ====>   Divide both sides by 9  ====>


7%5Ek%2A50 = 5%5Em%2A14  ====>


50%2A7%5Ek = 2%2A7%2A5%5Em  ====>  Comparing prime factors and their degrees and using the theorem on uniqueness 
                         of decomposition integers into the product of their prime factors


Answer.  m = 2,  k = 1.