If the eight-digit number 5a32465b is divisible by 88, then 2a + 5b =

For a number to be divisible by 88, it has to be divisible by both 11 & 8 .

Divisibility rule of 8 :  The number formed by last three digits of a given number has to be divisible by 8.
                 So, "65b" has to be a multiple of 8. 
                 This is only possible at b = 6. 

Divisibility rule of 11 : A number is divisible by 11 if the difference between the sum of the digits at the odd places and the sum of the digits at the even places is either 0 or a multiple of 11.
                    So, Sum of the digits at odd places = 5 + 3 + 4 + 5 = {17}.
                          Sum of the digits at even places = a + 2 + 6 + b = {a + b + 8}.
                                             The difference is |a+b-9| .

As we deduced 'b' is 6 : |a-3| which has to zero only, because it can't be 11 as the maximum value  'a' can take is a single digit number.
                                                       Therefore, a = 3.

Hence the value of 2a + 5b = 2(3) + 5(6) = 36.