* President says Republicans' math doesn't add up
* Warns on deep budget cuts, middle-class tax breaks
* Republicans have offered no complete fiscal plan
June 15 (Reuters) - Aiming to shoot holes in Republicans' budget plans, U.S. President Barack Obama is challenging reporters to verify his conclusions that they can't slash taxes and shrink deficits without gutting popular government programs and tax breaks.
Obama on Thursday charged that the numbers in Republican rival Mitt Romney's plan for taxing, spending and deficits have been only vaguely sketched out and their math does not add up.
The president said Romney and Republican lawmakers want to keep the tax cuts approved under President George W. Bush, "add another $5 trillion in tax cuts on top of that" and cut spending by $1 trillion, all while trimming the deficit.
"Now, I'm looking forward to the press following up and making sure that, you know, I'm not exaggerating," Obama said in a campaign speech in Cleveland, Ohio, a key battleground state.
Obama says that for Romney achieve his fiscal goals, he would have to drastically cut spending on programs that help the middle class and eliminate popular tax breaks such as deductions for mortgage interest deduction and exclusions for healthcare benefits.
The president warned that if cuts on the scale Romney laid out were enacted, then "10 million college students would lose an average of $1,000 each in financial aid; 200,000 children would lose the chance to get an early education in the Head Start program."
Fewer medical and scientific research grants would be available, millions would not benefit from the healthcare reform law Republicans want to repeal and there would be major changes to coverage for millions on Medicaid and Medicare, Obama said.
QUESTION ON REPUBLICAN PLAN
"The cuts to this part of the budget would be deeper than anything we've ever seen in modern times," Obama said. "This is not spin ... This is what they're presenting as their plan."
Except that Romney and the Republicans have not laid out a complete plan for what they want to do on taxes, the budget and the deficit.
Instead, they have made some general proposals, avoiding details on the hard parts, and preserving room for political maneuver, while largely focusing on attacking Obama in their campaign to unseat him in the Nov. 6 election.
The Romney campaign chafed at the suggestion that it has not provided enough details on its economic plan, forwarding press releases that include many of his planned cuts, including proposals to repeal Obama's healthcare overhaul and privatize the Amtrak passenger rail service.
"Governor Romney has repeatedly made clear that his tax proposal is revenue-neutral and that the lower rates would be offset by a broader base as well as economic growth," said campaign spokeswoman Andrea Saul.
Romney has said repeatedly that his tax policies would boost economic growth.
"I am absolutely convinced that you are going to see an extraordinary resurgence of America's economy. It's going to come roaring back with the right policy," Romney told supporters at a campaign fundraiser in Chicago on Thursday evening.
Although he has not offered a comprehensive plan, Romney has vowed to slash marginal tax rates for all Americans by 20 percent; scrap the alternative minimum tax and the estate tax; eliminate taxes on capital gains, interest and dividends for those earning less than $200,000; and cut the top corporate income tax rate to 25 percent from 35 percent while reducing corporate tax breaks.
But some analysts say there are still a lot of unanswered questions.
The Romney campaign wants the election to be a referendum on Obama's steering of the economic recovery, which has been uneven, highlighted by a dismal May jobs report.
"If you are going to make the entire premise of your campaign getting the economy back on its feet, it would be important to see some details," said Greg Valliere, an analyst for investors at Potomac Research Group. "His numbers don't add up.
"How can he pay for a new 20 percent tax cut while retaining all of the Bush tax provisions? He vaguely talks about deep spending cuts but where?" Valliere said.
Various budget-cutting proposals have been put forward by Republican lawmakers, including Representative Paul Ryan.
Obama appeared to get some of the figures he mentioned in his speech from Ryan's budget plan, which Romney has embraced, but which also is short on specifics about targeted programs.
The Tax Policy Center, a centrist think tank, estimated in March that the Romney plan would cut federal tax revenues by $480 billion in 2015, or just under $5 trillion over 10 years.
The center noted its analysis was incomplete because Romney has not spelled out how he would "broaden the base" of taxpayers to help lower rates. Romney has said that some tax benefits for the wealthy might have to be curtailed.
Obama also said 70 percent of the benefits from Romney's proposed tax cuts would go to those making more than $200,000 a year. A Tax Policy Center analysis shows 67 percent of Romney's proposed tax cuts would benefit those in upper income groups.