Thune: CLASS repeal dead on arrival in the Senate

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"I think we'd get a majority; I don't think we'd get 60," Thune said.

Still, Thune said, the House vote would create "momentum" in the Senate, where Democrats would be put on the spot. Republicans hope to embarrass Democrats by forcing them to support a voluntary payroll tax deduction program that the Obama administration itself has said is unsustainable.

Even getting a simple majority in the Senate would be a tall order. Republicans would need to persuade at least four Democrats to vote for repeal and break party unity, something they've failed to do in the House so far. Only four House Democrats — Reps. Jim MathesonJim MathesonWork begins on T infrastructure plan New president, new Congress, new opportunity First black GOP woman in Congress wins reelection MORE (Utah), Mike Ross (Ark.), John BarrowJohn BarrowDem files Ethics complaint on Benghazi panel Barrow thanks staff in farewell speech The best and the worst of the midterms MORE (Ga.) and Ron KindRon KindHouse GOP campaign arm targets Democrats over ObamaCare anniversary Here's how Congress can get people to live healthy lifestyles House Democrats identify vulnerable incumbents for 2018 cycle MORE (Wis.) — voted for repeal when it came up in committee, and the first three had voted against the reform law in the first place.