If both the President and Vice President can no longer serve, who becomes President?

Answer: Speaker of the House

Article 2, Section 1, Clause 6, Constitution of the United States: In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President, and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected.

In the case that the president can no longer serve, the vice president would serve as president.

If the vice president cannot serve, the line of succession falls to the speaker of the House, then to the Senate president pro tempore, then to Cabinet members.

    The Cabinet line of succession is:
    1. Secretary of State
    2. Secretary of the Treasury
    3. Secretary of Defense
    4. Attorney General
    5. Secretary of the Interior
    6. Secretary of Agriculture
    7. Secretary of Commerce
    8. Secretary of Labor
    9. Secretary of Health and Human Services
    10. Secretary of Housing and Urban Development
    11. Secretary of Transportation
    12. Secretary of Energy
    13. Secretary of Education
    14. Secretary of Veterans Affairs
    15. Secretary of Homeland Security

    Source: CNN

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