Recommended reading which deals with variations on this problem at length: Calendrical Calculations section 1.12, "Cycles of Years".

Distributing m (here, $num) "evenly" into n (here, scalar(@array)) elements, some will be int(m/n), and m%n will be 1 more than that. So, putting all the extras at the front gives:

my %result;
my $num = 13;
my @array = qw(a b c d e);
@result{@array} = map int($num/@array) + $_ < $num%@array, 0..$#array;
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To do it randomly, use @result{shuffle @array} instead.

Spreading them out as evenly as possible gives:

@result{@array} =
  map int($num/@array) + ($_ * ($num%@array) % @array < $num%@array),
    0..$#array;
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(See formula 1.57 in the book.)