Donald Trump won the Arizona primary Tuesday night, collecting all of the state’s 58 delegates and keeping him on track to win the GOP presidential nomination on a first ballot. Sen. Ted Cruz won the Utah caucuses and took all 40 delegates.

The Associated Press called the Arizona race at about 11:20 p.m. EDT and CNN promptly followed suit.

In early returns, with 46 percent of the vote counted, he had about 47 percent — a strong showing, but still far from the dominant showings that previous front-runners were racking up at this point in the race.

Mr. Cruz managed to get just 21 percent of the vote and Ohio Gov. John Kasich received 10 percent. Sen. Marco Rubio, who dropped out of the race last week after defeat in his home state of Florida, actually was outpolling Mr. Kasich with 17 percent of the vote, according to AP.

Mr. Cruz, the real-estate tycoon’s closest competitor, was hoping to score a comeback victory overnight in Utah. If Mr. Cruz is able to win a majority of the vote there, he would claim all of the state’s 40 delegates. Otherwise, the state would split its delegates among candidates proportionate to their tallies.

Mr. Cruz’s hopes of winning the state were being hampered by Mr. Kasich, the third major GOP candidate in the race, who has refused to drop out despite winning so far only once, in his home state of Ohio.

Mr. Kasich has no mathematical way of winning the nomination on a first-ballot vote at the July convention, but is hoping for a contested convention where delegates would shift to him.

For now, though, Mr. Trump remains in the driver’s seat, needing to win about 60 percent of the remaining available delegates to avoid a contested convention.

His victory in Arizona, which is a winner-take-all state regardless of whether the winner gets a majority or merely a plurality of the vote, kept him on track.

His win is also another vindication of his stiff immigration stance, which is popular among Republicans in the state. Mr. Trump had touted the support of, among others, Maricopa County Sheriff Joe Arpaio.

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