The Social Security full retirement age is 66 for most baby boomers, and you receive a smaller monthly payout if you sign up at a younger age. Some retirees sign up even later in order to qualify for larger monthly payments later in retirement. However, the Medicare eligibility age remains 65. So, if you want to wait until 66 or later to claim Social Security, you will have to sign up for Medicare separately at age 65. Here’s how to sign up for Medicare while delaying claiming Social Security.