What Republican president had a Democrat vice president?

That depends on how you care to look at it. Only one President has chosen a Vice President from a different political party, and as it so happens, it was indeed a Republican President who chose a Democrat to serve as his Vice President. During the 1864 election, Abraham Lincoln (a Republican) chose Andrew Johnson (a Democrat) to replace Hannibal Hamlin as Vice President. However, due to the Civil War being fought, the Republican party was temporarily "absorbed" into the National Union Party, and both Lincoln and Johnson were elected as members of that party. Upon the assassination of Abraham Lincoln, Johnson served his term as President without changing party affiliation. So, the question can be answered two ways. On the one hand, if you consider life-long political affiliations, then Abraham Lincoln was a Republican until the day he died, and Andrew Johnson was a Democrat his entire life (in 1869, he formally became a Democrat again). So, in that regard, the answer is "Abraham Lincoln". On the other hand, since normal political affiliations didn't apply to the 1864 election specifically, you can also contend the answer is "none" since technically speaking neither Lincoln nor Johnson were affiliated with their normal political parties during that one term.