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Hillary Clinton declared herself the Democratic Party's presidential nominee after a victory in the New Jersey primary, becoming the first woman to head a major party's presidential ticket.

"Thanks to you we've reached a milestone, the first time in our nation's history that a woman will be a major party's nominee," Clinton told supporters in New York, noting she was speaking in a hall with a glass ceiling and joking about breaking the glass ceiling of US politics.

Clinton's win in New Jersey lifted her above the threshold of 2,383 delegates needed to secure the party's presidential nomination.

Early Wednesday, US broadcasters CNN and NBC called a win for Clinton over rival Senator Bernie Sanders in the California primary - the delegate-rich state that is the biggest prize in the final six-state primary contests held on Tuesday.

Clinton won in the states of New Mexico and South Dakota as well, while Sanders came out on top in North Dakota and Montana.

The results marked the end of five-month primary process and both Clinton and presumptive Republican nominee Donald Trump turned their attention to a contest against each other in November elections.

Clinton said she wants to help write the next chapter in US history and take America forward, lambasting Trump for proposals she says would take the country backwards.

This election "really is about who we are as a nation," Clinton said. "It's about millions of Americans coming together and saying we are better than this, we won't let this happen in America."

She accused Trump of promising supporters an economy and a world he cannot recreate, saying the Republican "wants to win by stoking fears and rubbing salt in wounds and reminding us daily just how great he is."

Trump added to his own delegate count with victories in all five Republican contests in New Jersey, New Mexico, South Dakota, Montana and California.

The billionaire businessman, who as the last Republican candidate in the race already is the party's presumptive nominee, won large majorities in the states, according to official but incomplete results.

Trump told supporters he would "lead the Republican Party to victory" in November, stressing he would work hard to win the votes of Republicans who had supported other candidates in the primaries as well as Democrats and independent voters.

He specifically reached out to Sanders' supporters who feel alienated by their party leaders, saying "we welcome you with open arms."

Despite Clinton's declaration of victory and apparently insurmountable math, Sanders refused to concede the nomination, instead vowing to fight on.

Speaking to a raucous crowd of supporters in Santa Monica, California, Sanders said he would contest the June 12 primary in the nation's capital Washington, and "take our fight" to the party nominating convention in Philadelphia in July.

"I am pretty good at arithmetic and I know that the fight in front of us is a very, very steep fight," he said. "But we will continue to fight for every vote and every delegate we can win."

"We take our fight for social and economic justice, racial justice and economic justice to Philadelphia," he declared.

Clinton supporter Lareen Russell told dpa in Santa Monica that she voted for Clinton "because it is time for a woman in the White House."

Sanders "has some great ideas, but I think it is time for him to back off. Clinton has the experience," Russell said.

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