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