There are 200 Democrats and 233 Republicans in the House (2 seats are currently vacant). A majority of 217 is currently needed to pass legislation. All Democrats could be expected to vote for a clean continuing resolution to fund the government, like the one passed in the Senate. In other words, Speaker John Boehner could pass legislation to avoid a government shutdown with less than 20 Republican votes. The same is true for passing legislation to raise the debt ceiling and avoid the U.S. government defaulting on its obligations for the first time in history.
Think about that. Boehner only needs 20 Republican votes AND he has them. He just will not use them because he is afraid of the Tea Party and Tea Party funders like the Koch brothers.