Positive integers

Factor the left-hand side of the second equation: $2a(b+5)=3b+257$

Do the same thing to the other side: $2a(b+5)=3(b+5)+242$

Isolate the constant and simplify: $(2a-3)(b+5)=242$

The only "reachable" factors of 242 are 22 and 11. 

We can achieve these numbers when $a = 7$ and $b = 17$.

Thus, $ab = 17 \times 7 = \color{brown}\boxed{119}$

a + b  =   24   ⇒    b = 24  - a

So

2ab + 10a  = 3b + 257

2a (24 -a) + 10a  = 3(24 -a) + 257

48a  -2a^2 + 10a  = 72 - 3a + 257        rearrange as

2a^2 - 61a +  329    = 0         factors as

(2a  - 47) ( a - 7)    = 0         

The second factor set  = 0   gives an  integer for "a"

a - 7  =  0 

a =7

b = 24  - 7  =17

ab  =  (7)(17)   =   119