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 Given that a and b are positive integers and that a + b = 24 , what is the value of ab if 2ab + 10a  = 3b + 257 ?

 Apr 16, 2022
 #1
avatar+2437 
+1

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, \(ab = 17 \times 7 = \color{brown}\boxed{119}\)

 Apr 16, 2022
 #2
avatar+124594 
+1

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 

 

cool cool cool

 Apr 16, 2022

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