LAS VEGAS — When most people talk about divided government, they mean the gridlock in Washington where the Democratic president jousts with the Republican House. But the more interesting divide may well be between President Obama and the growing ranks of Republican governors who will help decide the fate of many of his policies.

Even as Mr. Obama was handily re-elected to a second term this month, Republicans won their 30th governor’s seat. Now, many of the same governors who tried and failed to block the president’s health care law in court are being charged with putting it into place. This week, as they gathered here for a meeting of the Republican Governors Association, they sought and won an extension of a Friday deadline to begin to decide how they will comply with the law.

Gov. Bob McDonnell of Virginia, who wrapped up his term as chairman of the group, noted that 60 percent of the states, with about 180 million people, now have Republican governors. He made it clear that they would sometimes try to check the president’s powers, mentioning the 10th Amendment, which reserves powers to the states.

“We will work with the president where our ideas can overlap; we’ll make a good-faith effort in all respects to do that,” Mr. McDonnell said. “But where there are unfunded mandates on the states or there is trampling of the 10th Amendment or undermining of federalism that hurts the state or costs us money, then we’ll fight it.”