Appearing on the "Late Show" with David Letterman, President Barack Obama scolded Mitt Romney over his caught-on-camera remarks, declaring that "if you want to be president, you gotta work for everybody, not just for some."

Letterman had asked Obama on Tuesday about Romney's remarks at a fundraiser several months ago in which the Republican standard-bearer essentially wrote off the president's core supporters as having a victim mentality and depending on government handouts.

"Is that what rich guys at country clubs are talking about?" Letterman asked.

"I don't know what he was referring to," the president said.

"But I can tell you this: When I won in 2008, 47 percent of the American people voted for John McCain," Obama said. "They didn't vote for me. And what I said on election night was: 'Even though you didn't vote for me, I hear your voices, and I'm going to work as hard as I can to be your president.'"

"And one of the things I've learned as president is you represent the entire country. And when I meet Republicans as I'm traveling around the country, they are hard-working family people who care deeply about this country. And my expectation is, is that if you want to be president, you gotta work for everybody, not just for some," Obama said.

Acknowledging his own 2008 gaffe in which he said Americans "bitter" about their lot in life "cling" to religion and gun rights, Obama said "I immediately said I regret this" because it "sent the wrong message to the country." (Not really. He first stood by them, then eventually worked his way to "I didn't say it as well as I should have." It was, not surprisingly, messy.)

"All of us make mistakes, all of us say the wrong thing once in a while," Obama told Letterman. "People understand you're going to make mistakes on the campaign trail. What I think people want to make sure of, though, is that you're not writing off a big chunk of the country."

"This is a big country. And people disagree a lot, but one thing I've never tried to do--and I think none of us can do in public office--is suggest that because someone doesn't agree with me that they're victims or they're unpatriotic."

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"There are not a lot of people out there who think they're victims," he said. "There are not a lot of people who think they're entitled to something."

Republicans have worked to paint the election as a battle between their plans for an "opportunity society" and what they deride as Obama's "entitlement society," which they claim is an approach in which government knows best. Romney made a variation on that argument in an interview Tuesday with Fox News Channel's Neil Cavuto.

"What I think the majority of people, Democrats and Republicans, believe is is that we've got some obligations to each other, and there's nothing wrong with us giving each other a helping hand," Obama said.

"I think that's a good investment for America, and that's—if you want to be president and you want to bring people together, I think that's the attitude that you've got to have."