Hillary Clinton claimed victory Tuesday in the New Jersey, New Mexico and South Dakota primaries as she claimed the Democratic Party’s presidential nomination and made clear she’s now preparing for the general election.

The results in New Jersey, New Mexico and South Dakota will have little bearing on the race, as the former first lady earlier this week secured the number of delegates needed to claim the nomination, according to an Associated Press count.

Her win in the Puerto Rico primary over the weekend and the support of party superdelegates gave her the needed 2,383 delegates.

But the victories still were important for the Clinton campaign, which is in danger of losing the California primary to Mr. Sanders in what would be an embarrassing loss in the nation’s largest state. The wins in New Jersey and New Mexico will blunt a possible loss in California.

In addition to New Jersey, New Mexico, South Dakota and California, Democrats in North Dakota and Montana also went to the polls Tuesday.

The result in New Jersey is of little surprise. While there had been scant polling in the state, the surveys that were conducted gave Mrs. Clinton a big advantage.

A recent CBS News/YouGov poll, for example, put Mrs. Clinton up by 27 points over Mr. Sanders. Surveys also had shown Mrs. Clinton ahead in New Mexico.

Despite the losses, Mr. Sanders and his supporters remain defiant. The senator has said he’ll keep campaigning until the party convention in July, even as leading Democrats on Capitol Hill sharpen their calls for him to drop out of the race and unite behind Mrs. Clinton.

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