When it comes to signing up for Social Security, you have plenty of choices to look at. You can file as early as age 62, or you can delay your claim until age 70 for the maximum monthly benefit you can get based on your personal earnings history. In fact, you can even file for Social Security after age 70, even though doing so doesn't make financial sense.

You may decide to file for Social Security at age 65 if you're planning to enroll in Medicare at that age. But claiming Social Security at 65 is a move you might regret.

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When you risk an income shortfall

You're allowed to start receiving Medicare coverage at age 65, and you can actually enroll a few months before your 65th birthday. If you've lost access to a workplace health plan, or you have a lousy plan through your employer, then signing up for Medicare at age 65 exactly could be a very smart move.

But one thing you should know is that you absolutely do not have to sign up for Social Security and Medicare at the same time. It's more than possible to be a Medicare enrollee for years before starting Social Security. And waiting to claim Social Security beyond age 65 could do a lot of good for your retirement finances.

You're entitled to the full monthly Social Security benefit your earnings history makes you eligible for at full retirement age (FRA), which is either 66, 67, or somewhere in between. So if you claim Social Security at 65, you'll automatically slash your monthly benefit to some degree.

Granted, that hit to your benefit won't be nearly as drastic as the hit you'd take by filing at age 62. But still, if you're able to hold off on claiming Social Security until FRA or beyond, it could very much pay to do so.

The savings you bring with you into retirement could end up getting depleted in your lifetime. Social Security, on the other hand, guarantees you your monthly benefit for life. So the higher that number is, the more financial stability you might enjoy. Plus, a delayed Social Security claim could do you a lot of good if you're low on savings.

How to sign up for Medicare alone

Because online enrollment for Medicare has you going to the Social Security Administration's website, you may be inclined to assume that if you're signing up for the former, you must also sign up for the latter. But rest assured that you will have the option to only enroll in Medicare if you're not ready to start getting Social Security benefits just yet.

Now if you're on Medicare without Social Security, you'll have to pay your monthly Part B premiums directly. Social Security enrollees have those premiums deducted from their monthly benefits. But that's really not a big deal, and Medicare offers choices for making those payments automatically.

So all told, if you decide that you'd rather not sign up for Social Security at age 65 because of the hit to your monthly benefits, don't. And you won't have to worry about it impacting your Medicare enrollment, either.