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Nicola Sturgeon said there would be a "widespread sense of relief" after Boris Johnson announced his resignation.
The prime minister has stepped down as party leader after losing the support of his ministers and MPs. More than 50 MPs resigned in a 24-hour period.
Speaking outside Number 10, Mr Johnson said he would continue as prime minister until his successor is chosen.
But the first minister questioned if this was sustainable.
Nicola Sturgeon told BBC Scotland: "The chaos and complete lack of integrity that has characterised Boris Johnson's premiership has in the last few days descended into complete and utter farce, and all at a time when people in every part of the UK are struggling with very real challenges.
"First and foremost there will be an overwhelming and very widespread sense of relief today that Boris Johnson's time as prime minister - which should probably never have been allowed to happen in the first place - is coming to an end."
She said it was incredible to suggest he would stay as prime minister for another three to four months.
'He will cause more chaos'
"It is just an unsustainable proposition to say he will continue to inhabit Number 10," she said. "I'm not sure anybody could look at Boris Johnson and conclude he is capable of genuinely behaving as a caretaker prime minister.
"He will want to do things and in the process of that undoubtedly cause even more chaos than he has already."
She said the "democratic deficit" inherent in the Westminster government would not be fixed with change of prime minister.
"The sooner he is out of Number 10 - and preferably that is today - the better," she added.
Ms Sturgeon said she did not believe a general election would be imminent.
"For Scotland, we are going to see in the next few months one prime minister that we did not vote for replaced by yet another prime minister we did not vote for - and would not vote for if given the chance," she said.
"For Scotland, the Westminster system doesn't represent our best interests and that is why we don't just need a change of prime minister or a wider changing of the guard at Westminster, Scotland needs a permanent alternative to Westminster which is why the independence choice is so necessary."
Earlier, Ms Sturgeon had said that although her differences with Boris Johnson were many and profound, she understood that leadership was difficult and brought with it many stresses and strains, and wished Mr Johnson and his family well on a personal level.
'Them's the breaks'
Mr Johnson had vowed to "keep going" following the wave of resignations despite senior members of his Cabinet, including new chancellor Nadhim Zahawi, urging him to resign and "leave with dignity".
On Thursday morning, it emerged that Mr Johnson had decided to step down.
During his resignation speech in Downing Street, he said: "It is clearly now the will of the parliamentary Conservative Party that there should be a new leader of that party and therefore a new prime minister.
"And I've agreed with Sir Graham Brady, the chairman of our backbench MPs, that the process of choosing that new leader should begin now and the timetable will be announced next week.
"And I've today appointed a Cabinet to serve, as I will, until a new leader is in place."
Mr Johnson addressed the British public, telling them: "I want you to know how sad I am to be giving up the best job in the world, but them's the breaks."
Scottish Conservatives leader Douglas Ross said the decision to resign was the right one.
He told BBC Scotland: "I think it was clear when you lose the chancellor of the exchequer, when the health secretary resigned, there was a very clear movement that the prime minister could not continue.
"But I think the last 48 hours have made it far more difficult."
He said Mr Johnson should not remain in his role until a replacement was appointed.
"I believe the best option would be to have an interim prime minister, " he said.
Mr Ross added: "What we need to do now is focus on the issues that matter to people. The public just want politicians to work for them and get down to the jobs that will improve their lives."
Replacement will be 'complicit'
Scottish Labour leader Anas Sarwar said: "The entire country will be breathing a sigh of relief that Boris Johnson is going, but let's not forget that whoever replaces him is complicit.
"This is a party that is rotten to the core, that allowed a cheat, a charlatan, a man devoid of any principles or any values to become the prime minister of this country - and they inflicted his carnage on this country for years.
"That's why not only do we need to see the back of Boris Johnson, we also need to see the back of this Tory government.
"And that's why we need an election to elect a Labour government that can bring the values of decency, integrity and principles back into our country."
Former Scottish Tory leader Baroness Ruth Davidson said the idea that the prime minister was going to stay on until the party conference was "arrant nonsense".
Scottish Greens co-leader Patrick Harvie said it was a relief that Boris Johnson's "disastrous tenure as prime minister" was coming to an end.
He said: "Let's not kid ourselves: Johnson was never fit to hold high office.
"Those in the Conservative Party who facilitated Johnson's lying, his lawbreaking, and his incompetence must be held to account too.
"Suggestions that Johnson can remain in office until October are ridiculous. He needs to go, and he needs to go now."
The Scottish Liberal Democrats tweeted: "It has always been clear that Boris Johnson is unfit to lead our country. The public won't forgive the Conservatives for propping him up so long.
"Lifelong conservative voters are fed up with being taken for granted. Trust in the Conservatives has gone and it won't be coming back."