You can find out for free if you want. That's how I know mine. I probably won't bother checking again.
I love it when people assume stupid things. There are plenty of free (no annual fee) credit cards about. If you pay them off each month in time, you also don't pay interest. But it's still a loan which gets noted as such in your credit score - ie you get a record for good payment without having to pay extra for it. The loan is for the period between the transaction happening and the payment being made, ie a month or so.
Neither do I. In fact, I get a small cash cut of the transactions which go through my CC. The CC company make their money through the merchant fees.