Given that a and b are positive integers and that a + b = 24 , what is the value of ab if 2ab + 10a = 3b + 257 ?
+2668 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, \(ab = 17 \times 7 = \color{brown}\boxed{119}\)
+130071 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
