60 seats in the Senate would gives party or majority the right to invoke 'cloture'. This limits debate and brings legislation to a vote.
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Cloture on limiting debate and changing Senate rules.
60 votes limits debate (3/5ths)
67 votes changes Senate rules (2/3rds)
"In 1975, the Senate revised its cloture rule so that three-fifths of sworn senators (60 votes out of 100) could limit debate, except for changing Senate rules which still requires a two-thirds majority of those present and voting to invoke cloture "
https://en.m.wikipedia.org/wiki/Filibuster_in_the_United_States_Senate
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Clôture means to bring something to a vote without further debate or a filibuster.
"Clôture is French for 'the act of terminating something'.
"It was introduced into the Parliament of the United Kingdom by William Edwart Gladstone to overcome the obstructionism of the Irish Parliamentary Party and was made permanent in 1887.
"It was subsequently adopted by the United States Senate and other legislatures.
"The name cloture remains in the United States; in Commonwealth countries it is usually closure or, informally, guillotine; in the United Kingdom closure and guillotine are distinct motions."
https://en.m.wikipedia.org/wiki/Cloture
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Supermajority - required to change Senate rules.
"While a supermajority vote can technically be specified as any fraction or percentage greater than one-half ( 50%), commonly used supermajorities include three-fifths (60%), two-thirds (67%), and three-quarters (75%)"
More actions requiring a supermajority:
https://www.thoughtco.com/the-supermajority-vote-in-us-government-3322045
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How many votes to pass a bill with everyone present:
https://fair.org/uncategorized/how-many-votes-does-it-take-to-pass-a-senate-bill/
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Cloture Rules
What is cloture? Why did Senate come up with 60 as the number for cloture?
"Currently, the only Senate procedure that can overcome a filibuster is cloture. Cloture requires the support of three-fifths of the Senate or a minimum of 60 votes, except on proposals to change rules, which require two-thirds of Senators to give their approval. Also, cloture cannot be voted on until two days after it is proposed. If the Senate invokes cloture on a bill, amendment, or other issue, consideration of the matter is limited to 30 additional hours during which each Senator may speak for no more than one hour. Cloture does not stop debate immediately; it only ensures that debate cannot continue indefinitely."
http://www.washingtonpost.com/wp-srv/interactivity/congress-question.html
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Nuclear option - what this option allows the Senate to pass legislation with a simple majority of 51 votes that would otherwise take 60 votes to pass.
"In November 2013, Senate Democrats led by Harry Reid used the nuclear option to eliminate the 60-vote rule on executive branch nominations and federal judicial appointments, but not for the Supreme Court. In April 2017, Senate Republicans led by Mitch McConnell extended the nuclear option to Supreme Court and the nomination of Neil Gorsuch ending the debate. As of January 2018, a three-fifths majority vote is still required to end debates on legislation."
https://en.m.wikipedia.org/wiki/Nuclear_option
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More in depth
https://www.senate.gov/CRSpubs/577d2a5e-2b47-4045-95fa-a76398e41461.pdf
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Can a bill pass with less than 51 votes if all the Senators are not present and accounted for? So far I have not been able to find a definitive answer on this.