Sen. John McCain won a hard-fought South Carolina primary Saturday night, avenging a bitter personal defeat in a bastion of conservatism and gaining ground in an unpredictable race for the Republican presidential nomination. Democrats Hillary Rodham Clinton and Barack Obama split the spoils in Nevada caucuses marred by late charges of dirty politics.

“We’ve got a long way to go,” McCain told The Associated Press in an interview. He quickly predicted that his victory in the first southern primary would help him next week when Florida votes, and again on Feb. 5 when more than two dozen states hold primaries and caucuses.

“This is one step on a long journey,” Clinton told cheering supporters in Las Vegas. She captured the popular vote, but Obama edged her out for national convention delegates at stake, taking 13 to her 12.

Obama issued a statement that said he had conducted an “honest, uplifting campaign … that appealed to people’s hopes instead of their fears.”

If the Democrats had co-front-runners, the Republicans had none, and looked to the first southern state primary to begin winnowing an unwieldy field.

McCain defeated former Arkansas Gov. Mike Huckabee in a close race in the state that snuffed out his presidential hopes eight years ago. The Arizonan was gaining 33 percent of the vote to just under 30 percent for his closest rival.

“It just took us a while. That’s all. Eight years is not a long time,” McCain told the AP.

Former Tennessee Sen. Fred Thompson was in a struggle for third place with about 16 percent, after saying he needed a strong showing to sustain his candidacy. Another Republican, California Rep. Duncan Hunter, dropped out even before the votes were tallied.

Interviews with South Carolina voters leaving their polling places indicated that McCain, an Arizona senator, and Huckabee, a former Arkansas governor, were dividing the Republican vote evenly. As was his custom, McCain was winning the votes of self-described independents.

South Carolina was the second half of a campaign double-header for Republicans.

Former Massachusetts Gov. Mitt Romney cruised to victory earlier in the day in the little-contested Nevada caucuses.

No matter the state, the economy was the top issue in all three races on the ballot.

Republicans in Nevada and South Carolina cited immigration as their second most-important concern.

Among Democrats in Nevada, health care was the second most-important issue followed by the Iraq war, which has dominated the race for months.

With three contests on the ballot, it was the busiest day of the presidential campaign to date, and fittingly enough for a pair of wide-open races, every contest produced a different winner.

Romney rolled to victory in Nevada Republican caucuses, winning roughly 50 percent of the vote in a multi-candidate field.

With a black man and a woman as the leading contenders, the Democratic race was history in the making — and increasingly testy, as well.

Before the votes were tallied, Obama was critical of former President Clinton, telling reporters, “It’s hard to say what his intentions are. But I will say that he seems to be making a habit of mischaracterizing what I say.”

Obama’s campaign manager, David Plouffe, issued a written statement accused the Clinton campaign of “an entire week’s worth of false, divisive attacks designed to mislead caucus-goers and discredit the caucus itself.”

Clinton declined to comment on the allegation.

Whatever the hard feelings, she told supporters they would fade by the fall general election campaign. “We will all be united in November,” she said, as the crowd chanted “HRC, HRC.”

Her campaign issued a statement citing numerous reports of voter intimidation. It also accused UNITE HERE, a union supporting Obama, of running a radio commercial that was “one of the most scurrilous smears in recent memory.” The ad, broadcast in Spanish, said Clinton “does not respect our people” and called her shameless.

Interviews with Democratic caucus-goers indicated that Clinton fashioned her victory by winning about half the votes cast by whites, and two-thirds support from Hispanics, many members of a Culinary Workers Union that had endorsed Obama. He won about 80 percent of the black vote.

Obama had pinned his Nevada hopes on an outpouring of support from the 60,000-member union. But it appeared that turnout was lighter than expected at nine caucuses established along the Las Vegas Strip, and some attending held signs reading, “I support my union. I support Hillary.”

Democrats looked next to South Carolina to choose between Obama, the most viable black candidate in history, and Clinton, seeking to become the first woman to occupy the White House. The state is home to thousands of black voters, who are expected to comprise as much as half the Democratic electorate.

After that, the race goes national, with more than 20 states holding primaries and caucuses on Feb. 5 and 1,678 national Democratic convention delegates at stake.

The split Democratic verdict in Nevada resulted from the proportional manner in which delegates were awarded. Obama emerged with one more than Clinton because he ran strongly in rural areas.

Overall, Clinton leads the delegates race with 236, including separately chosen party and elected officials known as superdelegates. Obama has a total of 136, and former North Carolina Sen. John Edwards has 50.

Romney struck first on the day among the Republicans.

The former Massachusetts governor learned of his Nevada victory when his wife Ann announced it on the public address system of his chartered jet. “Keep ’em coming. Keep ’em coming,” he said.

Romney had campaigned for months to win early contests in Iowa and New Hampshire, and his candidacy was in trouble when he lost both. He retooled his appeal to the voters in the days leading to the Michigan primary, though, focusing on the economy and trumpeting his experience as a businessman.

En route to Florida, he presented reporters with his ambitious economic stimulus plan, $233 billion in all. It includes tax rebates as well as tax cuts for individuals, as well as tax cuts for businesses.

Nevada Republicans said the economy and illegal immigration were their top concerns, according to a survey of voters entering the caucuses. Romney led among voters who cited both issues.

Mormons gave Romney about half his votes. He is hoping to become the first member of his faith to win the White House. Alone among the Republican contenders, Rep. Ron Paul of Texas aired television ads in Nevada.

Nearly complete returns showed Romney winning more than 50 percent of the vote, with Paul and McCain far behind vying for second. Thompson and Huckabee trailed.

Romney also won at least 17 of the 31 Republican National Convention delegates at stake. McCain and Paul won at least four apiece, while Thompson and Huckabee each won two. Hunter and Rudy Giuliani each won one delegate — the first of the campaign for the former New York mayor.

Nevada offered more delegates — 31 versus 24 — but far less appeal to the Republican candidates than South Carolina, a primary that has gone to the party’s eventual nominee every four years since 1980.

That made it a magnet for Thompson, who staked his candidacy on a strong showing, as well as for Romney, McCain and Huckabee.

McCain, a former Vietnam prisoner of war, appealed to a large population of military veterans in South Carolina, and stressed his determination to rein in federal spending as he worked to avenge a bitter defeat in the 2000 primary.

Huckabee reached out to evangelical Christian voters, hoping to rebound from a string of disappointing showings since his victory in the Jan. 3 Iowa caucuses.

Romney campaigned on a pledge to help restore the state’s economy, much as he did in winning Michigan.

In South Carolina, the economy and immigration were cited as top issues, with more than half the voters saying illegal immigrants should be deported. Conservatives and white evangelical voters turned out in heavy numbers, according to the polling place interviews.

Survey data in both states were from polls conducted for The Associated Press and the television networks by Edison Media Research and Mitofsky International.

South Carolina primary voters coped with equipment difficulty and bad weather. Election officials in the area around Myrtle Beach brought out paper ballots after some electronic voting machines failed to work properly. Snow fell in the northern part of the state, which has little snow removal equipment.