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# [Brachylog], 16 bytes

    g₂;.z∋zbᵗ×ᵐ/ℤ↰|w

[Try it online!][TIO-ks7xdgx9]

This is a function submission, which inputs *n* from the left and *A* from the right (as a list of two-element lists). Output is to standard output.

## Explanation

    g₂;.z∋zbᵗ×ᵐ/ℤ↰|w
    g₂                  place two list wrappers around the left input
      ;.                append the right input
        z               cycling zip; because the left input is length 1,
                          this pairs it with each element of the right input
         ∋              find {the first} element for which no assertions fail
    [at this point, the element looks like [[n], [num, denom]]
     for some pair [num, denom] in A]
          z             rearrange to [[n, num], [n, denom]]
           bᵗ           remove first of last pair ([[n, num], [denom]])
             ×ᵐ         take product of each inner list ([n×num, denom])
               /ℤ       divide; assert result (n×num÷denom) is an integer
                 ↰      recurse (loop back to start of program)
                  |     if all else fails (i.e. ∋ found no elements)
                   w    output the current value to standard output

[Brachylog]: https://github.com/JCumin/Brachylog
[TIO-ks7xdgx9]: https://tio.run/##SypKTM6ozMlPN/r/P/1RU5O1XtWjju6qpIdbpx@e/nDrBP1HLUsetW2oKf//39DA0szQ2NTc2Ph/dLS5hY6paaxOtKmOMZA01AGxDQ11jMBC5rGxAA "Brachylog – Try It Online"