WASHINGTON — Mitt Romney has narrowly won the Michigan Republican presidential primary. The former Massachusetts governor defeated Rick Santorum, who had waged a hard-fought battle for the state. Michigan is critical for Romney because it is the first of the nation’s industrial battleground states to vote in the GOP nominating race — and it is also his home state.

Earlier Tuesday night, Romney easily won the Arizona GOP primary.

Two other candidates, Newt Gingrich and Ron Paul, made little effort in either state, pointing instead to next week’s 10-state collection of Super Tuesday primaries and caucuses.

Romney’s Arizona triumph came in a race that was scarcely contested, and he pocketed all of the 29 Republican National Convention delegates at stake in the winner-take-all race.

Michigan was as different as could be – a hard-fought and expensive battle in Romney’s home state that he could ill afford to lose and Santorum made every effort to win.

Returns from 62 percent of Michigan’s precincts showed Romney at 40 percent and Santorum at 36 percent. Paul was winning 11 percent of the vote to 7 percent for Gingrich.

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In Michigan, 30 delegates were apportioned according to the popular vote. Two were set aside for the winner of each of the state’s 14 congressional districts. The remaining two delegates were likely to be divided between the top finishers in the statewide vote.

With his victory in Arizona, Romney had 152 delegates, according to The AP’s count,, compared to 72 for Santorum, 32 for Gingrich and 19 for Paul. It takes 1,144 to win the nomination at the Republican National Convention in Tampa next summer.

In interviews as they left their polling places, Michigan voters expressed a notable lack of enthusiasm about their choices. Just 45 percent said they strongly favored the candidate they voted for, while 38 percent expressed reservations and 15 percent said they made the choice they did because they disliked the alternatives.

The lengthening GOP nomination struggle has coincided with a rise in Democratic President Barack Obama’s prospects for a new term. A survey released during the day showed consumer confidence at the highest level in a year, and other polls show an increase in Americans saying they believe the country is on the right track.

Along with the improving economy, the long and increasingly harsh campaign, in which Gingrich and Santorum have challenged Romney as insufficiently conservative, has prompted some officials to express concern about the party’s chances of defeating Obama in the fall.

Exit polling showed a plurality of Republican voters in both Michigan and Arizona saying the most important factor to them in the primaries was that a candidate be able to beat Obama in November. Romney won that group in Michigan, where it mattered most, and also prevailed among voters in the state who said experience was the quality that mattered most.

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Santorum ran particularly well among voters who cited a desire for strong conservatism or strong moral character.

The polls surveyed both primary day and absentee or early voters. Interviews were conducted at 30 polling places in each state. Early results from Arizona’s poll included interviews with 1,617 voters, including 601 absentee or early voters interviewed by phone. In Michigan it was 2,133 interviews including 412 absentee or early voters interviewed by telephone. The margin of sampling error for both polls was plus or minus 4 percentage points.

Not even the opening of polls on Tuesday brought an end to the squabbling between the two leading Republicans.

Romney accused Santorum of trying to hijack a victory in Michigan by courting Democratic votes through automated telephone calls and suggested his rival was appealing to conservatives by making the kind of “incendiary” statements he would not.

“I’m not willing to light my hair on fire to try and get support,” Romney said. “I am what I am.”

Santorum brushed aside the allegations of hijacking, saying Romney had appealed for support from independents in earlier states.

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“We’re going to get voters that we need to be able to win this election. And we’re going to do that here in Michigan today,” Santorum said, referring to blue collar voters with a history of swinging between the parties.

The exit poll said about 10 percent of the day’s Michigan primary voters were Democrats.

If nothing else, the unexpected clash on Romney’s home field dramatized that two months into the campaign season – after nearly a dozen primaries and caucuses – the GOP race to pick an opponent for President Barack Obama remains unpredictable.

Unopposed for renomination, Obama timed a campaign-style appearance before United Auto Workers Union members in Washington for the same day as the Michigan primary. Attacking Republicans, he said assertions that union members profited from a taxpayer-paid rescue of the auto industry in 2008 are a “load of you know what.”All of the Republicans running for the White House opposed the bailout, but even in the party’s Michigan primary a survey of voters leaving polling places showed about four in ten supported it.

Michigan loomed as a key test for Romney as he struggled to reclaim his early standing as front-runner in the race. The first of the industrial battleground states to vote in the nominating campaign, it is also the place where the former Massachusetts governor was born and where he won a primary when he first ran for the party nomination four years ago.

But Santorum, a former Pennsylvania senator, rolled into the state on the strength of surprising victories on Feb. 7 in caucuses in Minnesota and Colorado plus a non-binding primary in Missouri. He quickly sought to stitch together the same coalition of conservatives and tea party activists that carried him to a narrow victory in the Iowa caucuses that opened the campaign nearly two months ago.

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The Michigan primary was open to Republicans or any voter who declared they were Republican for the purpose of voting, and there was precedent for an influx of outsiders influencing the outcome.

A dozen years ago, John McCain defeated the heavily favored George W. Bush, relying on the support of Michigan independents and Democrats. Exit polls then showed that Bush won 66 percent of Republican votes, while McCain won 82 percent of self-described Democrats and 67 percent of independents. Together, the non-Republican voters accounted for more than half the electorate.

In a measure this year of the state’s importance to the battle for the nomination, the two leading candidates and the super PACs that support them spent about $6 million on television advertisements, and Romney and Santorum spent much of the past 10 days crisscrossing the state in search of support.

Arizona was Romney’s to lose, judging by the behavior of his rivals, who spent little time campaigning in the state and no money advertising on its television airwaves.

In all, there were 59 delegates at stake in the two states.

Arizona awarded all 29 of its delegates to the winner of the statewide vote.

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In Michigan, by contrast, 30 delegates were apportioned according to the popular vote. Two were set aside for the winner of each of the state’s 14 congressional districts. The remaining two delegates were likely to be divided between the top finishers in the statewide vote.

Romney entered the night the delegate leader in The Associated Press count.

He had 123, compared to 72 for Santorum, 32 for Gingrich and 19 for Paul. It takes 1,144 to win the nomination at the Republican National Convention in Tampa next summer.

There are 40 delegates at stake in Washington caucuses on Saturday, followed by 419 on Super Tuesday, including big state primaries in Ohio and Georgia.

Already, the television advertising wars were under way. Romney and restore Our Future, the super PAC that supports him, have spent more than $3 million combined on ads in Ohio.

In Michigan, Santorum campaigned heavily for the support of tea party activists and other non-establishment Republicans, appearing in churches at times and often dwelling on social issues, as is his custom. In a string of attention-gathering remarks in the race’s final days, he said Obama was a snob who wanted everyone to attend college, said he nearly threw up over a speech that candidate John F. Kennedy gave in 1960 about the separation of church and state, and said Romney was uniquely unqualified to defeat Obama because the two men shared so much in common on issues like health care.

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The former Massachusetts governor made a play for tea party support, too, at a pair of appearances, but for the most part campaiged on his pledge to use his background as a successful businessman to help create jobs and fix the economy. Last week, he issued a call for 20 percent across-the-board cuts in personal income tax rates.

But he was hampered by off-the-cuff comments that reinforced his difficulty in reaching out to struggling voters in a state with 9.3 percent unemployment. He said at one point that his wife drives a couple of Cadillacs, and at another that he was friends with some of the owners of NASCAR teams.

At a rare news conference after the polls opened on Tuesday, he conceded that his own mistakes had hurt his campaign.

The primaries in Michigan and Arizona were the first contests since Romney squeaked out a victory over Paul in the Maine caucuses on Feb. 11, a lull of two and a half weeks.

Except for a debate in Arizona last Wednesday and a brief burst of campaigning in the hours before and after, Romney and Santorum have focused their time and campaign money on Michigan.

Polls showed Santorum racing to a large advantage after his victories on Feb. 7, before the weight of attack ads by a Romney-aligned super PAC and the candidate himself began to narrow and finally erase the gap in many surveys.

In the end, the combination of Romney and the outside group accounted for about $3.8 million in TV ads, compared to about $2.2 million for Santorum and a super PAC supporting him.

While that gave Romney an advantage, it wasn’t nearly as lopsided as in some of the earlier states.