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UPDATE 3-Mexicans turn to church as earthquake death toll hits 320

(Adds students returning to school, new death toll)

MEXICO CITY, Sept 24 (Reuters) - Mexicans packed churches on Sunday to pray for the victims of the country's deadliest quake in 32 years as rescue teams searched against the odds for any survivors trapped under rubble since Tuesday's tremor shook Mexico City and nearby states.

As another aftershock jolted southwestern Mexico on Sunday, the death toll from Tuesday's 7.1 magnitude earthquake climbed to 320 people. With thousands of buildings damaged, survivors slept on the street outside their homes and estimates of the cost of the earthquake ran as high as $8 billion.

Many have been traumatized by the second major quake to strike Mexico City in their lifetime after a devastating 1985 tremor killed an estimated 10,000 people.

In the Basilica of Our Lady of Guadalupe, the national shrine of the majority Roman Catholic country, thousands of people gathered to pray.

"I came to ask God for strength for those who lost loved ones and for the Virgin to watch over us and keep us safe," said 69-year-old Maria Gema Ortiz. "Thanks to all those who came from other countries to help. Thanks to all and long live Mexico!"

Makeshift places of worship have popped up next to the crumbling cement and mangled steel of collapsed buildings in the deeply religious country.

In upscale Roma, one of the hardest-hit neighborhoods of the capital, a priest led mass for nearly two dozen people under a blue tarp while a nun handed out small cards with an image of the Virgin of Guadalupe, who according to the Catholic faith first appeared to an Aztec convert in 1531.

More than 44,000 public schools in six states were due to reopen on Monday, but only 103 of the 4,000 public schools in Mexico City would open so as not to impede rescue and relief efforts.

In addition, the National Autonomous University of Mexico, with 350,000 students at campuses in and around Mexico City, will resume classes on Monday.

Like many Mexicans, 36-year-old Claudia Avila was determined to return to some semblance of normality.

"We are afraid, but life must go on," said Avila, whose sons are 9 and 16. "Tomorrow I will take my children back to school. They know that if something happens, they must protect themselves. It has been a rude awakening."

Rescuers narrowed their search to a handful of buildings in the sprawling metropolitan area of 20 million people, using advanced audio equipment to detect signs of life beneath tonnes of rubble, with help from teams from as far afield as Israel and Japan.

"The search and rescue in Mexico City continues as a priority, with cooperation from national and international groups," tweeted Miguel Angel Mancera, Mexico City's mayor.

The search for survivors continued in a ruined office building in the Roma neighborhood and in a five-story apartment building in historic Tlalpan.

Authorities called off efforts in the upper-middle class Linda Vista zone after pulling 10 bodies from the rubble, while work at the Tlalpan building was briefly halted on Saturday by a magnitude 6.2 aftershock.

SEARCH FOR SURVIVORS

Roberto Hernandez, 62, leading a group of Mexico's famed "mole" rescue workers at the collapsed office building in Roma, said he believed 30 people were trapped in the rubble, though it was not clear how many were sill alive.

"We can't guarantee there is life but we can guarantee we'll turn over every last stone," Hernandez said.

Tuesday's quake, which flattened dozens of structures in Mexico City, was the second major earthquake to strike the country of 127 million people this month.

A massive 8.1 magnitude quake on Sept. 7 off the southwestern coast of Mexico killed around 100 people, most of them in the nearby states of Oaxaca and Chiapas. A series of aftershocks since then have sown panic.

The latest tremor, of 5.7 magnitude, struck on Sunday off the west coast, with its epicenter 80 km (50 miles) south-southwest of Tonala, in Chiapas, the U.S. Geological Survey said. There were no immediate reports of significant damage.

Many more quakes are likely, warned Xyoli Perez Campos, director of Mexico's National Seismological Service.

"We have already recorded more than 4,300 aftershocks," Campos said. "So more aftershocks are to come. What we don't know is if they are going to be of significant magnitude."

President Enrique Pena Nieto visited 12 communities in Mexico state, which borders the capital city, promising that aid would be directed only to those truly affected by the quake and that the government would help rebuild homes and businesses.

"I am publicly promising that ... the affected homes, the families affected and the people in the whole of Joquicingo and the state of Mexico, get back on their feet," he said from that town.

(Additional Reporting by Michael O'Boyle; Writing by Anthony Esposito and Daniel Flynn; Editing by Daniel Trotta and Sandra Maler)