(CNN)Donald Trump cruised to victory in the South Carolina Republican primary and Florida Sen. Marco Rubio and Texas Sen. Ted Cruz are battling to be the chief Republican challenger to him. Hillary Clinton also defeated Vermont Sen. Bernie Sanders in Saturday's other contest, the Nevada Democratic caucuses.
Winners and losers from South Carolina, Nevada make their pitch on CNN
All five of those candidates appeared on CNN's "State of the Union" Sunday morning to give their take on the night's events.
Hillary Clinton, coming off a hard-fought victory in the Nevada caucuses on Saturday, appeared exclusively on CNN and acknowledged she has work to do in convincing independent voters and others that she's trustworthy.
"I understand that voters have questions -- I'm going to do my very best to answer those questions. I think there's an underlying question that maybe is really in the back of people's minds and that is, 'Is she in it for us or is she in it for herself?' " Clinton said.
"I think that's a question that people are trying to sort through. And I'm going to demonstrate that I've always been the same person, I've always been fighting for the same values, fighting to make a difference in people's lives, long before I was in elected office, even before my husband was in the presidency," she said.
Asked if she has a problem she needs to address among Latino voters because entrance polls show her losing them to Sanders, Clinton said the actual voting tallies paint a different picture.
"That's just not what our analysis shows," she said. "We don't believe the so-called entry polls are particularly accurate. If you look at the precincts, if you look at where we dominated, there's a lot of evidence we did well with every group of voter."
Donald Trump's general election prediction: He'll face Hillary Clinton, and the two will bring out "the greatest turnout in history."
"Frankly, if she gets indicted, that's the only way she's going to be stopped. I think it's going to be Hillary and myself," the Republican real estate mogul said.
Trump's comments came the morning after he cruised to victory in South Carolina's primary -- giving him two wins and one second-place finish in the first three GOP contests.
He laid out his own road map to general election victory, pinpointing two states -- Michigan and New York -- that he said he'd sweep into the Republican column.
"I'll win states that aren't in play. I'll win states that Republicans don't even think of," Trump said.
And he predicted he'd earn a "tremendous amount" of support from African-Americans.
"I'm going to do great with the African-Americans. African-American youth is 58% unemployed. African-Americans in their prime are substantially worse off than the whites in their prime, and it's a very sad situation," he said.
For Trump, Saturday's South Carolina victory was an important one in stunting challenges from top-tier rivals like Texas Sen. Ted Cruz -- but also because it knocked his foil, former Florida Gov. Jeb Bush, out of the race.
He also clarified a comment he made last week that seemed to suggest he supported an individual mandate for health insurance.
Marco Rubio is portraying himself as the lone candidate who can unite Republicans now that the party's presidential field is narrowing.
"I give us the best chance to unify," the Florida senator said.
He said the GOP race is increasingly about "who can win."
"Who do the Democrats fear most? Who do they not want to run against? I think everyone now acknowledges that's me," he said. "We've got to bring the Republican Party together. We're not gonna win -- we're not going to beat Hillary Clinton or Bernie Sanders -- if we're still divided in September or October."
While Donald Trump won South Carolina's primary Saturday, Rubio was buoyed by a slim second-place finish over Texas Sen. Ted Cruz -- and perhaps even more by a key rival, former Florida Gov. Jeb Bush, deciding to drop out after that primary.
He said the race has a "different dynamic now," with three real contenders -- Trump, Cruz and himself.
Ted Cruz argued that since he's the only candidate who has been able to beat Donald Trump, he's the candidate conservatives should turn to as the chief alternative to the businessman.
"It is now apparent that the only campaign that can beat Donald Trump and that has beaten Donald Trump is our campaign," he said in an interview on "State of the Union" with CNN's Jake Tapper.
Cruz dismissed the idea that he underperformed in South Carolina, a state with a strong religious and evangelical population -- both attributes that should favor him. And he downplayed Trump's victory, arguing that his lead narrowed in recent weeks and the margin of victory should have been greater.
Cruz also swiped that Rubio should have done better in South Carolina and tried to lump the policy proposals of Donald Trump in with Hillary Clinton's.
Bernie Sanders brushed off his Saturday Nevada loss to Hillary Clinton, saying that just a few months his campaign was still considered a "fringe campaign" that would never have come within striking distance there.
"The truth is that for a campaign that started out as a fringe campaign at 3% in the polls we have enormous momentum," Sanders told CNN's Jake Tapper on Sunday's "State of the Union." "You have noticed that one of the recent national polls actually had us ahead of Hillary Clinton, in state after state her margin is narrowing. So I think people are responding to our message of a rigged economy where ordinary Americans work longer hours for lower wages and almost all new income goes to the top one percent."
Sanders lost Nevada by about five points. He played up his success among Latino voters there, but acknowledged he need to do better among black voters.
"I believe we won the Latino vote, which is a huge, huge way forward for us. We did badly with the African-American vote," Sanders said.