Billionaire businessman Donald Trump won the Republican primary in the southern US state of South Carolina Saturday, vowing to pick up momentum as the field of candidates narrowed with former Florida governor Jeb Bush quitting the race.

Trump had 32.6 per cent of the vote, state election officials said. His closest rivals Florida Senator Marco Rubio and Texas Senator Ted Cruz battled for second place, with 22.5 per cent and 22.1 per cent of the vote respectively.

The results give Trump momentum heading into the next contest in Nevada on Tuesday, and the Super Tuesday primary on March 1 when more than a dozen states hold primaries.

Meanwhile, former secretary of state Hillary Rodham Clinton beat Vermont Senator Bernie Sanders in a separate Democratic caucus in the western state of Nevada.

Trump told supporters that he anticipates big victories ahead.

"Let's have a big win in Nevada," he said. "Let's have a big win in the [March 1 primaries]. Let's put this thing away and let's make America great again."

Trump supporters had expressed anger at a broken political system and the influence of big campaign donors in politics.

Trump "can't be bought and he'll tell you to your face what he thinks and how he feels," Michael Reilly, 18, told dpa as he voted for the first time.

The candidates trailing Trump hope to emerge as the alternative to the real estate tycoon, whose brash campaign style has equally alienated and energized Republican voters. The first two contests in Iowa and New Hampshire saw wins for Cruz and Trump, respectively.

But Trump dismissed pundits who say as that as the Republican field narrows, the anti-Trump vote will consolidate.

"What they don't understand is that as people drop out, I'm going to get those votes too," he said.

Jeb Bush, the son and brother of former US presidents, ended his quest to be the Republican nominee for president after failing to make a strong showing in South Carolina and the two previous states to vote.

Announcing his decision to supporters, the former Florida governor said he's had a "front row seat" to the US presidency and anyone assuming that office must know he or she is "a servant, not a master."

The next president must serve with "honour and decency" said Bush, who had taken on Trump over his divisive rhetoric and profane language.

Both Rubio and Cruz sought to cast their results as providing momentum to their campaigns moving forward.

"Now, practically speaking, it's down to three," Rubio said. "And I know that our campaign gives us the best chance not just to come together, not just to unify our party but to unify our country and to grow this movement."

Cruz insisted that he was the only candidate who had beaten Trump in Iowa and could beat him again in the future.

In Nevada, with 88 per cent of the ballots counted, Clinton had won 52.6 per cent of the vote to 47.3 per cent for Sanders.

"Some may have doubted us, but we never doubted each other," Clinton said in a victory speech.

Saying her campaign was about building "ladders of opportunity," Clinton took a subtle dig at her opponent by noting that the US was not "a single-issue country."

Sanders's campaign has focused sharply on economic inequality and the power of large financial institutions.

Sanders congratulated Clinton and praised his supporters for making the race close after being so far down in the polls only weeks before.

Clinton had once held a substantial lead bolstered by the western state's large Hispanic population, but saw her advantage slip after Sanders notched a large victory in New Hampshire and ended in a near tie in Iowa.

The Clinton campaign had stressed she was better positioned to gain support of the minority voters who are a key Democratic voting bloc in Nevada.

Her campaign had described Nevada, South Carolina and southern states with primaries next month as her "firewall" on the path to the nomination, as Sanders, a self-described democratic socialist, has seen his message of economic equality resonate with voters.

The Democrats next hold their own primaries in South Carolina on February 27 and Republican hold caucuses in Nevada on Tuesday.

Latest news

Syrian opposition rules out future role for President al-Assad

The Syrian opposition said Friday it would not accept any role for President Bashar al-Assad in the future of the war-torn country, reacting to a recent US shift saying that removing al-Assad is no longer a priority for Washington.

Russian Army integrates breakaway forces of Georgian province

Parts of the small fighting forces of the Georgian breakaway province of South Ossetia have been placed under Russian military control, Russian Defence Minister Sergei Shoigu said on Friday.

Czech Republic's Pilsner Urquell beer is now Japanese

Japanese brewing company Asahi completed its takeover of the Czech brewery Pilsner Urquell on Friday, Asahi said in a statement.

Judge approves 25-million-dollar settlement of Trump University case

A US district judge on Friday approved a 25-million-dollar settlement of lawsuits and state fraud allegations against Trump University, the US president's now-defunct business venture.

Former Thai premier Thaksin to junta on reconciliation: 'Cut me out'

Former Thai prime minister Thaksin Shinawatra on Friday announced that he is not interested in the junta-led reconciliation process, three days after the junta handed him a half-a-billion-dollar tax bill for his past business deal.

Dalic: We welcome possible deal between Agrokor and banks

The government welcomes the possibility of an agreement being concluded between the Agrokor food company and creditor banks, and the bill on vitally important companies is not a fallback plan but the result of the government's care for the overall economic and financial stability of Croatia, Deputy Prime Minister and Minister of Economy Martina Dalic told a press conference in Zagreb on Friday.

Croatia, China sign action plan for cooperation in agriculture

The Croatian and Chinese ministries of agriculture on Friday signed an action plan for cooperation in the field of agriculture for the period 2017-2018, the Croatian ministry said in a statement.

ZSE indices up, Agrokor shares in focus of investor interest

The Zagreb Stock Exchange (ZSE) indices on Friday rose by more than 1.8%, with stocks of the Agrokor food and retail concern being in the focus of investor interest again.

Berlin police defend handling of Berlin market attacker

Berlin police defended themselves on Friday against accusations that they stopped surveillance on Berlin Christmas market attacker despite knowing in June 2016 he was dangerous.

Croatia, creditors tailor emergency measures to save tottering giant

Croatia's tottering retail and food giant Agrokor reached an agreement with its creditors, putting its debts standby and allowing it to continue working during emergency restructuring, the Croatian branch of Austria's Erste Bank said Friday.

Agrokor's creditors say standstill agreement to go into force today

A standstill agreement regarding the Agrokor concern's existing financial obligations to banks will take effect on Friday, additional capital will be injected into the concern in the coming days and the concern will be actively restructured, which includes a change of its management, it was said on Friday after a meeting between Agrokor's suppliers and creditor banks.

Palestinians, UN slam Israel's new settlement plan

Palestinians, Israeli activists and the UN lambasted the government of Israeli Prime Minister Benjamin Netanyahu on Friday, a day after it gave the go-ahead for the first new West Bank settlement in a quarter of a century.

South Sudan rebels release three abducted foreign oil workers

South Sudanese rebels have released three foreign engineers they abducted in early March in the oil-rich Upper Nile region, Foreign Affairs Ministry official Mawein Makol Arik said on Friday.

Turkish opposition: Imprisoned party chief has gone on hunger strike

The head of Turkey's pro-Kurdish opposition party has launched a hunger strike from prison.

European leagues threaten Champions League schedule clashes

The European Professional Football Leagues (EPFL) on Friday threatened schedule clashes on Champions League matchdays in an ongoing dispute with the governing body UEFA.

Danish court revokes citizenship of IS volunteer

A Danish appellate court on Friday stripped a man of his Danish citizenship for volunteering to fight for the extremist Islamic State in Syria.

Banks and Agrokor agree on key elements of standstill agreement

Member banks of the coordinating committee of financial creditors and representatives of the Agrokor food company have in principle agreed on key elements of a standstill agreement, which is expected to be signed later today, announcing changes in the company's management team, Erste Bank said in a statement on Friday afternoon.

Syrian man on trial in Sweden; mosque attack labelled terrorism

A Syrian man went on trial Friday in the southern Swedish city of Malmo, charged with terrorism and arson after an attack last year on a building used as an assembly hall by Shiite Muslims.