Repeat this GCD operation
Jelly, 9 bytes

ÆEz0Ṣ€ZÆẸ


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How it works

ÆEz0Ṣ€ZÆẸ  Main link. Argument: A (array)

ÆE         For each n in A, compute the exponents of n's prime factorization.
           E.g., 2000000 = 2⁷3⁰5⁶ gets mapped to [7, 0, 6].
  z0       Zip 0; append 0's to shorter exponent arrays to pad them to the same
           length, then read the resulting matrix by columns.
    Ṣ€     Sort the resulting arrays (exponents that correspond to the same prime).
      Z    Zip; read the resulting matrix by columns, re-grouping the exponents by
           the integers they represent.
       ÆẸ  Unexponents; map the exponent arrays back to integers.