Seven different positive integers have a mean of 12 and a median of 10. What is the largest possible value of the greatest of them?

Total $=84$. Make all the others as small as allowed: below the median 1, 2, 3; the median 10; above it 11 and 12 (they must exceed 10 and differ). Greatest $=84-6-10-23=45$. (47 wrongly lets a value above the median equal 10.)