One explanation: Those at the base of his support are the same who question the president's legitimacy -- and playing to that fringe has been Trump's ticket to success so far.

He stoked those inaccurate beliefs on NBC's "Meet the Press," when he was asked whether he'd be comfortable with a Muslim president.

"Some people have said it already happened, frankly," Trump responded in a clear reference to Obama. "But of course you won't agree with that."

"I don't talk about people's faith. Now, in all fairness, he said he was a Christian and he said he is a Christian. He attended the church of Rev. Wright. And so, you know, I'm willing to take him at his word for that. I have no problem with that," he said.

He again dodged the question on during an interview with George Stephanopoulos on ABC's "This Week."

Asked whether Obama is a Muslim, Trump said, "George, you have raised the question. I haven't raised the question. I don't talk about it and I don't like talking about somebody else's faith. He talks about his faith and he can do that. But I don't talk about other people's faith. It's not appropriate for me to talk about somebody else's faith."

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Obama's faith is back in the headlines after Trump failed to correct a supporter at a New Hampshire town hall last week who said Obama was born outside the United States and is a Muslim. Trump's non-response and his dance around the issue on Sunday suggests the real estate mogul doesn't want to cross supporters who share these discredited beliefs.

Trump has long fueled questions about Obama's background.

Four years ago, Trump was the driving force behind demands for Obama's birth certificate. This year, he has courted Sarah Palin, the former Alaska governor and darling of the anti-Obama right, whose political rallies in 2008 featured similar sentiments.

Recent polling suggests misperceptions about Obama's beliefs and background -- he is a Christian and was born in Hawaii -- continue to exist.

A CNN/ORC poll found this month that 20% of Americans say they are at least suspicious Obama was born outside the United States. Nearly 3-in-10 Americans surveyed said they believe Obama is a Muslim -- including 43% of Republicans and 54% of Trump's supporters.

Trump has been critical of John McCain, who in 2008 took a microphone away from a supporter who made similar comments and corrected her.

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Democratic presidential front-runner Hillary Clinton cast Trump's handling of his supporters' rhetoric in stark terms Sunday on CBS' "Face the Nation" -- saying Trump should have handled his campaign rally attendee just as McCain did seven years ago.

"He is fueling a level of paranoia and prejudice against all kinds of people, and when you light those fires, you better recognize that they can get out of control. And he should start dampening them down and putting them out," Clinton said.

"If he wants to talk about what he would do as president, that's obviously fair game," she said. "But to play into some of the worst impulses that people have these days, that are really being lit up by the Internet and other conspiracy-minded theories, is just irresponsible. It's appalling."

Clinton herself, though, was criticized for saying in 2008 Obama isn't a Muslim "as far as I know."

Trump wasn't the only Republican to court controversy over religion on Sunday.

Ben Carson didn't question Obama's faith or heritage but he said the United States should not elect a Muslim president. Still, he said he might vote for a Muslim for Congress.

"I would not advocate that we put a Muslim in charge of this nation. I absolutely would not agree with that," the retired neurosurgeon and Republican presidential candidate said on "Meet the Press."

Carson was asked Sunday whether a president's faith should matter to voters.

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"I guess it depends on what that faith is," he said. "If it's inconsistent with the values and principles of America, then of course it should matter. But if it fits within the realm of America and consistent with the Constitution, no problem."

Asked whether Islam is consistent with the Constitution, Carson said: "No, I don't -- I do not."

The views expressed by Trump and Carson are far from out of the GOP's mainstream. A new CNN/ORC poll shows that Trump remains at the front of the pack nationally, with 24% support in the crowded 16-candidate field. Just one percentage point behind Carly Fiorina, Carson is third, at 14%.

Other Republicans sought to avoid new controversy over Obama's beliefs and nationality.

An exasperated Florida Sen. Marco Rubio, another GOP presidential contender, said that he's "tired of this" and called it "a big waste of time" in an appearance on ABC's "This Week."

"This is -- of course -- he's born in the United States. He's a Christian. He's the president of the United States for the next year and a half, and we're going to move on," Rubio said of Obama.

He also called the questions raised by Trump and his supporters a distraction.

RELATED: Ben Carson: U.S. shouldn't elect a Muslim president

"Every time that we discuss these sorts of things, we're not talking about the family who's out there trying to make it -- they don't know if they're going to make it to Friday before a check bounces because they don't have enough money in their checking account," said Rubio, who moved up to fourth place at 11% support in the latest CNN poll.

Kentucky Sen. Rand Paul, who's courting the libertarian vote in the 2016 GOP race, called it "kind of crazy" to suggest Obama is a Muslim, and criticized Trump for failing to push back against his supporter, during an appearance on CBS' "Face the Nation."

"We shouldn't question the President's faith, and I think that's kind of crazy. And if someone does, we should, I think, rebuke that," he said.

However, he said he understands Carson's sentiments about a Muslim president.

"I just think that it's hard for us -- we were attacked by people who are all Muslim," Paul said.

He said the United States is unlikely to elect a Muslim president anytime soon largely due to demographics.

"It's not so much what religion you are, it's what you stand for. But I don't think that we're really anywhere near, probably, that happening -- because they're a small minority in our population," Paul said. "But the hard part is, while we are a very pluralistic society and we're open to all religions, more free than any other country, the problem we have is the people who have been attacking us have been all of one religion and so it's hard to separate that."