The Scottish Government put forward proposals at the end of previous year in an attempt to create a bespoke Brexit deal for Scotland - where the vote to remain was 62% - but Ms Sturgeon said they were "dismissed out of hand by the Prime Minister".
Founded in 1934, initially pushing for a devolution of powers to Scotland.
The election of Alex Salmond to the leadership in 1990 proved significant.
"The SNP has won the council elections".
The Scottish National Party (SNP) have returned the most councillors in the recent council elections held on Thursday 5 May.
"Every day feels hard, but there is no greater privilege than having the chance to make a difference in your country, particularly by being the leader of your government", she said.
The party now has 19 seats overall, one more than the Conservatives who now have 18 councillors after gaining seven seats.
Scottish Labour leader Kezia Dugdale said: "Nicola Sturgeon is trying to use Brexit as the excuse for another divisive referendum, but won't be straight with people about her position on European Union membership".
The move is in the face of her party's long-standing opposition to the policy, brought in by the SNP Government in 2011.
Ms Sturgeon and Mr Gethins visited the University's centre for Designer Quantum Materials before taking part in a discussion on the Erasmus programme with students who had studied overseas.
Mrs May suggested Ms Sturgeon and her party had a "tunnel vision" when thinking about another independence vote.
Tory MSP Donald Cameron said: "As far as running the country goes, the last 10 years is more about what the SNP hasn't achieved".
"We haven't got to that stage yet", she said.
The currency was seen as a major weakness of the pro-independence campaign in 2014, after the SNP's proposals for a currency union with rest of the UK were rejected by The UK Government.
The SNP objective will be to consolidate the spectacular gains made at the 2015 election.
Answering questions for the public, the Prime Minister was told by during one question she would have his vote "for a generation" if she stops Scotland from having another independence referendum.
Ms Dugdale promised that every Scottish Labour MP elected to Westminster would "oppose austerity and fight for the best Brexit deal for Britain".
Davidson's Conservative party, which holds just one seat in Scotland, has campaigned nearly exclusively on one issue - keeping the nation as part of the United Kingdom.
Where do they stand on Brexit?
"Yet still Nicola Sturgeon doesn't seem to have any coherent position on this".
The June 23 Brexit referendum called the future of the United Kingdom into question because England and Wales voted to leave but Scotland and Northern Ireland voted to stay.
However, Ms Sturgeon accepted that it was "possible, not necessarily desirable" that for "a period" an independent Scotland would be in the European Free Trade Association (Efta) and European Economic Area (EEA).
"Strengthening Scotland's hand in Brexit negotiations by giving the SNP a clear democratic mandate to demand a place for Scotland at the top table - with our continuing place in the single market a fundamental pillar of the UK's negotiating remit". The restriction of tax credits for any new claimants to a maximum of two children unless a woman can demonstrate a third or subsequent child was a result of rape.