Simplify a Date

Pyth - 116 87 bytes

jm++@rR3c."ayÖèÈÒ+J÷z4wëÝ~ñ!Õ¥´{mCØóy|å²z¼qP?ë"\qhd;ed.u/RYNPiFKsMcz\/K

Test Suite.

Jelly, 59 bytes

ṣ”/VµÆDf/2ị:@µÐĿị1¦€“£ṢtẒ⁽ẹ½MḊxɲȧėAṅ ɓaṾ¥D¹ṀẏD8÷ṬØ»ṣ⁶¤j€⁶j⁷

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How it works

ṣ”/VµÆDf/2ị:@µÐĿị1¦€“...»ṣ⁶¤j€⁶j⁷  Main link. Argument: mm/dd

ṣ”/                                Split at slashes.
   V                               Eval each chunk, yielding [m, d] (integers).
    µ                              Begin a new, monadic chain. Argument: [m, d]
             µÐĿ                   Execute the chain to the left until the results
                                   are no longer unique. Yield the list of all
                                   intermediate results.
     ÆD                              Compute the divisors of each number.
       f/                            Intersect them.
         2ị                          Select the one at index 2. If there is only
                                     one divisor, ị wraps around and selects 1.
           :@                        Divide [m, d] by this common divisor.
                        ¤            Combine the links to the left into a chain.
                 “...»                 Yield the month's name, space-separated.
                      ṣ⁶               Split at spaces.
                €                    For each pair...
             ị                          index into the month's names...
              1¦                        for the first element.
                         j⁶€         Join each pair, separating by spaces.
                            j⁷       Join, separating by linefeeds.