A Formula for Congruences
Haskell, 61 55 53 bytes

f x=[[p|p&lt;-[0,product x`div`n..],p`mod`n==1]!!0|n&lt;-x]


Defines a function f that takes input and gives output as a list of integers.

f x=[                                          |n&lt;-x]  (1)
              product x                                (2)
                       `div`n                          (3)


First we loop over all integers in the input (1). Then we take the product of all of the integers (2) and divide by n to get just the product of the non-n integers, which is P (3).

           [0,               ..]                       (4)
     [p|p&lt;-                     ,p`mod`n==1]           (5)
                                            !!0        (6)


Then we use the result (P) as the step value for a range starting at zero (4). We take the result, [0, P, 2P, 3P, ...], and filter it on values for which the result of a mod-n operation is one (5). Finally, we take the first element, which works thanks to lazy evaluation (6).

Thanks to @xnor for 2 bytes!