$\large { 2 }^{ a }+{ 2 }^{ b }+{ 2 }^{ c }+{ 2 }^{ 13 }={ 2 }^{ d }$

Given that $a,b,c,d$ are positive integers that satisfy the above equation.Then what is the maximum value of $a+b+c+d$ ?

The answer is 58.

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We know that ${ 2 }^{ x }+{ 2 }^{ x }=2^{ x+1 }$ From the problem ${ 2 }^{ a }+{ 2 }^{ b }+{ 2 }^{ c }+{ 2 }^{ 13 }={ 2 }^{ d }$ for the maximum value of a, b, c, d you should write in ${ 2 }^{ 13 }+{ 2 }^{ a }+{ 2 }^{ b }+{ 2 }^{ c }={ 2 }^{ d }$ where a = 13, b = 14, c = 15 and d = 16 so a+b+c+d = 13+14+15+16 = 58