sending letters using the Arduino Bluetooth Module

Hi,
I am making a project, and I need to send letters (or numbers) from my smartphone to my Arduino using the HC-05 bluetooth module. I am using App Inventor and the app was tested by a competent person who sait everything was right. Then I made the code in Arduino: no syntax errors. But when I press buttons on my app and send data (letters: a, b, c...) nothing answers.
Here's the arduino code:

char state = 121;

void setup() {
  pinMode(3, OUTPUT);
  pinMode(4, OUTPUT);
  pinMode(5, OUTPUT);
}

void loop() {

if(Serial.available() > 0){ 
    state = Serial.read(); 
 }

  if (state == 97) {
    digitalWrite(3, HIGH);
  }
  if (state == 98) {
    digitalWrite(3, LOW);
  }
  if (state == 99) {
    digitalWrite(4, HIGH);
  }
  if (state == 100) {
    digitalWrite(4, LOW);
  }
}

I tried changing the if(state == 97) to if(state == 'a') but it is still unresponsive.
Can you tell me where it is wrong?

Can you tell me where it is wrong?

You have not told us how the HC-05 is connected to the Arduino or whether it is successfully paired to the phone. So, no, we can't.

the app was tested by a competent person who sait everything was right.

Tested how?

In general, you should NOT have the HC-05 connected to the hardware serial pins. You need (obviously) those for debugging.

As I saw in a tutorial, I wired it on a Mega using pins TX and RX (pins 0 and 1). Obviously, pin RX connects to the TX and the TX connects to the RX. And yes, the device paired correctly. This leads me to think that the problem is in the code.

As I saw in a tutorial, I wired it on a Mega using pins TX and RX (pins 0 and 1).

Get it off of those two pins. Put it on TX1 and RX1, and use Serial1 to read from the device, IF there is anything to read. Use Serial, then, to debug your program.

You did connect at least two other wires to the HC-05, didn't you?