using pullstime

OK. Let's plunge on as you don't seem interested as to why it does not work

When you do

variableA = variableB % x

then variableA is assigned the remainder of variableB divided by the value of x

Change your code to add 2 extra print lines like this

      Serial.println(buttonPushCounter);
      Serial.print("buttonPushCounter % 5 = ");
      Serial.println(buttonPushCounter % 5);

Upload the code and press the button several times and post the output here