Centers for Medicaid and Medicare Services
Medicare is a Federal program which is the same nationwide. Re: Medicaid, there are significant variations among the States.
In 1964, his Great Society program introduced legislation which created Medicare and Medicaid.
Medicaid, would be a non-contributory program. That is because people can't afford it, just like food stamps. Medicare is a contributory program.
The US Centers for Medicaid and Medicare Services regulate the Medicare program, but the paperwork is often outsourced to private contractors.
The Medicare number is 1.800.MEDICARE. Medicaid is a State program; each State has its own contacts.
"Should" is a matter of opinion. Medicare is not an entitlement program (unlike Medicaid). It is an insurance program funded by payroll taxes on employees and employers.
Centers for Medicare and Medicaid Services, US Department of Health and Human Services
Lyndon B Johnson (1965)
Lyndon B. Johnson
Medicaid and Medicare
Social Security