Output Number Thinking

Algebra solution:

According to instructions, we get the equation

$\begin{aligned} 3(10a+b)-4(a+b)-2ab&=10a+b\\ \end{aligned}$

Then rearrange it, we get

$b=\frac{8a}{a+1}$

So the output value $\text{O}$ can be $14,36,77$ .

Therefore, the sum of the answer is $\boxed{127}$ .

Coding solution:

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for N in range(10,100):
  a=int((N-(N%10))/10)
  b=N%10
  S=a+b
  P=a*b
  O=3*N-4*S-2*P;
  if(N==O):
    print(N)
#The output is 14,36,77.

$N=\overline{ab}$ $\implies N=10a+b\;$ Following the given instructions, $N=10a+b\; {\color{#3D99F6}\longrightarrow}\; 30a+3b\; {\color{#3D99F6}\longrightarrow}\; 30a+3b-4(a+b) \; {\color{#3D99F6}\longrightarrow}\; 30a+3b-4(a+b)-2(ab)=O$ If $N=O$ , $\begin{aligned} 10a+b&=30a+3b-4a-4b-2ab \\ 0&=16a-2b-2ab \\ b(a+1)&=8a \implies b=\frac{8a}{a+1} \\ \end{aligned}$ Since $a$ and $a+1$ do not share a common factor (two consecutive numbers are always co-prime) and $b$ is an integer, $a+1$ must divide $8$ .

Hence, $a+1\in \{2,4,8\}\implies a\in \{1,3,7\}\implies (a,b)\in \{(1,4),(3,6),(7,7)\}\implies N\in \{14,36,77\}$

Therefore, the sum of all possible values of $O$ is $14+36+77=\boxed{127}$