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On any credit card (unless you are in a 0% APR period) always pay in full before the due date if you can. (Whether you pay before or after statement cuts is a detail, just pay before the due date) You don't want to pay unnecessary interest.
If you can't PIF, pay as much as you can. People warn about paying the miniumum for several months, because that does suggest financial weakness, but also maximizes the interest you are paying.
But as was also mentioned, technically Amex and other issuers CAN close your card for any or no reason. They won't, only if they have cause to believe there is a risk.
Paying in full is almost always the best option. In terms of scoring, you gain nothing by carrying a balance.
Amex will not close your account merely because you carry a balance. "Amex hates balances" is an overhyped mantra around here IMO.
What they do hate, as any lender does, is risky behavior. My limit is 15k. If I carry a $300 balance for 2 months, that's 2% utilization. Someone with that kind of utilization is hardly going to be considered a risk. Now if the balance were 13,000 instead, now I'm at 86% utilization. And if I started throwing some bare minimum payments in there, I could see them getting nervous. See the difference? Everyday, mild to moderate balance carrying is not going to cause you any trouble.
Amex or any lender can close your account at any time for any reason. So spending time being paranoid about it is unproductive. Just use your cards, PIF whenever you can or pay huge chunks if not (and reevaluate your budget if PIF is constantly not an option). You'll be fine.
Is it a charge card or a revolving credit card? Charge cards are supposed to be PIF every month. Revolving credit cards are allowed to carry a balance. If it's a revolving credit card then you should be okay. Best of luck!!!