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By Jonathan Martin, New York Times—

Hillary Clinton and Donald J. Trump overwhelmed their rivals in the Arizona primaries on Tuesday, according to The Associated Press, a show of might from two presidential front-runners who are hoping to avoid prolonging the nominating contest and begin training their fire on each other.

But Senator Bernie Sanders thrashed Mrs. Clinton in the Idaho and Utah Democratic caucuses, demonstrating his enduring appeal among liberal activists even as she closes in on the party’s nomination. And Senator Ted Cruz, who won the Republican contest in Utah, captured more than 50 percent of the vote, giving him all 40 of the state’s delegates and sustaining hope among Mr. Trump’s opponents that he can be slowed, if not stopped.

Mrs. Clinton’s commanding victory in Arizona, where 75 Democratic delegates were at stake, gave her the night’s biggest prize, and her margin there was substantial enough that Mr. Sanders was unlikely to emerge with significantly more delegates, though he took two states to her one.

Speaking to supporters in Seattle after winning Arizona, Mrs. Clinton looked past her primary. She used her remarks mainly to address the terrorist assault on Brussels and turned toward an attack on Mr. Trump and Mr. Cruz, the two leading Republican contenders.

“The last thing we need, my friends, are leaders who incite more fear,” Mrs. Clinton said. “In the face of terror, America doesn’t panic. We don’t build walls or turn our backs on our allies. We can’t throw out everything we know about what works and what doesn’t and start torturing.”

She added, “This is a time for America to lead, not cower, and we will lead, and we will defeat terrorism and defend our friends and allies.”

Mr. Trump easily defeated Mr. Cruz in Arizona, taking all 58 of its Republican delegates and adding to his delegate lead despite Mr. Cruz’s romp in Utah.

 

The victories recorded by Mrs. Clinton and Mr. Trump showcased the strengths that have propelled them to huge advantages in their respective nomination fights. Mrs. Clinton once again demonstrated her loyal following among older and nonwhite Democrats, both significant constituencies in Arizona. And Mr. Trump proved his appeal among immigration hard-liners, who make up a large bloc of Republicans in the border state.

But Tuesday has also shown why the two dominant candidates are still locked in primaries even as blossoms bloom and Easter approaches. Mrs. Clinton continues to struggle in heavily white states that vote through caucuses, like Idaho and Utah. And Mr. Trump’s landslide loss in heavily Mormon Utah was a reminder of his challenges with some of the Republican Party’s most religious voters, and of the uneasiness of many conservatives with their front-runner.

Still, Mrs. Clinton’s triumph in Arizona, which Mr. Sanders aggressively contested, offered a psychological boost as she heads into a stretch of contests in states likely to favor Mr. Sanders, like Alaska and Washington, where losses could underscore her lingering vulnerabilities among Democrats.

A hoarse Mr. Sanders, speaking to thousands of supporters in San Diego midway through the evening, claimed credit for what he called “record-breaking turnouts.” Ignoring his lopsided loss in Arizona, he noted that he had won 10 contests so far and predicted, correctly as it turned out, that he would win “a couple more tonight.”

Republicans hoping to stop Mr. Trump suffered another blow as he carried Arizona by a wide margin: He was on a course to receive more votes than Mr. Cruz and Gov. John Kasich of Ohio combined. If his opponents fail to defeat him in Wisconsin, where voters go to the polls in two weeks, they are unlikely to stop him from clinching the nomination on the last day of voting in June.

Mr. Trump’s easy victory in Arizona also provided a sharp rebuttal to assertions by Mr. Cruz that he would struggle in the remaining contests because so many of them allow only Republicans to vote. Aided by Arizona’s generous early-voting laws, Mr. Trump showed that he can win handily in states with closed Republican primaries, where Democrats and independents are barred from voting.

Mr. Trump’s more precise vulnerability appears to have been in states holding caucuses, where organizational strength can be decisive. But after Tuesday, there are no such contests left.

Turnout by voters in Arizona, Utah and Idaho was unusually high, with long lines — some snaking for several blocks — at polling places and caucus sites.

The ballot counting in Arizona was delayed as officials extended voting to account for people who were waiting in line, for hours in some cases, when the polls closed at 7 p.m. In Utah, some Democratic caucus sites had to print additional ballots to accommodate the turnout.

Tuesday’s Western contests came as Mr. Trump and Mrs. Clinton have both demonstrated strength in a string of recent primaries.

Mr. Trump, who won four of the five contests on March 15, including Florida and Illinois, has built a substantial delegate advantage over Mr. Cruz, and he campaigned aggressively in Arizona in the hopes of capitalizing on his success and reinforcing the perception that his nomination is inevitable.

He drew thousands of supporters last weekend to events near Phoenix and in Tucson, both of which drew impassioned protests.

An important part of what has galvanized his admirers and opponents alike is Mr. Trump’s tough talk on immigration. His harsh stance on the subject drove some of Arizona’s most prominent immigration hard-liners, including Sheriff Joe Arpaio of Maricopa County, the state’s largest, to Mr. Trump’s side.

The Democratic contenders sought to exploit the alliance between the sheriff and Mr. Trump, hoping to appeal to white liberals and to Hispanics, who make up around 30 percent of Arizona’s population.

“When I see people like Sheriff Arpaio and others who are treating fellow human beings with such disrespect, such contempt, it just makes my heart sink,” Mrs. Clinton said at a rally on Monday in Phoenix.

Mr. Sanders has been just as outspoken about Sheriff Arpaio, who has drawn criticism for his aggressive tactics in arresting and detaining illegal immigrants. “It’s easy for bullies like Sheriff Joe Arpaio to pick on people who have no power,” Mr. Sanders said to supporters on Monday in Flagstaff.

Arizona was the most heavily contested of the three states voting on Tuesday in the Democratic race, in which Mrs. Clinton has opened a nearly insurmountable lead after sweeping all five states that voted on March 15.

Only registered Democrats were allowed to vote in Arizona, posing an obstacle to Mr. Sanders, who typically overwhelms Mrs. Clinton among independents.

Utah and Idaho, by contrast, were voting through caucuses, not primaries, and have largely white populations — two frequent indications of success for Mr. Sanders.

But with Democrats allocating their delegates on a proportional basis, Mr. Sanders’s share of the combined 64 delegates offered by Utah and Idaho will do little to dent Mrs. Clinton’s lead of 319 pledged delegates heading into Tuesday’s votes.

On the Republican side, Mr. Trump enjoyed an advantage in Arizona thanks to the state’s expansive early-voting laws: Republicans were eligible to begin casting ballots, in person or through the mail, 26 days before the primary — a period in which Mr. Trump racked up a series of victories. And by the time polls opened Tuesday morning, more than half of Arizona Republicans had cast their ballots.

In a sign of how many Republicans had already voted, with 80 percent of Arizona precincts reporting, Mr. Kasich had received fewer votes than Senator Marco Rubio, who withdrew from the race last Tuesday.