A relative of Qabha told the Israel daily Haaretz that Qabha was a house painter and was returning home from the city of Jenin, where he had gone to buy paint, when his auto hit the soldiers.
Friday's incident happened as clashes between Palestinians and Israeli security forces erupted in various places in the occupied West Bank. One remains in critical condition.
The comment came as donor countries and others have worked to raise money for the beleaguered Strip, which United Nations officials say is facing a crippling shortage of clean water among other ills.
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It is still unclear whether the incident was a vehicle ramming attack or a mere traffic incident.
26-year-old Ala Qabha of the village of Barta'a near Jenin confessed to carrying out the attack, the Shin Bet security service said. "We will not rest until we bring all those who cooperate with terror to justice, we will not allow terror to become an accepted norm", President Reuven Rivlin stated. "This is not the first and not the last accident, and it happens every day".
The Palestinian terror group Hamas praised the attack, saying that it "proves our people's readiness to continue the Jerusalem intifada".
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The high will be near 66 with an east southeast wind of 15 miles per hour , gusting as high as 25 miles per hour . Tuesday night, the low will be around 31 with partly cloudy skies, according to the weather service.
A second tunnel, located in central Gaza, also was attacked "as part of the systematic effort to thwart the underground project from the Gaza Strip and in response to the recent incidents near the border fence".
The spokesperson added that the army used a new technique to destroy the tunnel that would prevent Hamas from being able to use it again, but would not elaborate on its nature.
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The coastal territory, which is now home to more than 2 million Palestinians, has been under an Israeli-led siege for more than a decade.