We prime factor 3888

                3888
                /  \
               2   1944
                   /  \
                  2   972
                      / \
                     2  486
                        / \
                       2  243
                          / \
                         3  81
                            / \ 
                           3  27
                              / \
                             3   9
                                / \
                               3   3 

So 3888 = 2435

To be a complete square, a positive integer must 
either be an even power of a prime, or
expressible as the product of even powers of primes. 
3888 is expressible as the product of an even power 
of 2 and an odd power of 3.  So the smallest number
we could multiply 3888 by to get a complete square
is 3, so that we would have 2436,
which is a product of even powers of primes 2 and 3.

Answer: 2

Checking: 3888×3 = 7776 = 1082

Edwin