Another solution without using any module:

#!/usr/bin/perl -w
use strict;

while (1) {
    print "Enter year (yyyy) and month (mm), separated by a space:";
    chomp(my $input = <STDIN>);
    last if !$input;

    my ($year, $month) = split / /, $input;
    my $m= $month % 12 + 1;
    my $y= $year+int($month/12);
    if ( $m<3) {
        $m+=12;
        --$y;
    }
    ++$m;
    my $dayname = (qw(Sat Sun Mon Tue Wed Thu Fri))[(int($m * 30.6) + 
+int($y * 365.25)-int($y / 100)+int($y / 400)) % 7];

    print "The last day of $month/$year will be a $dayname\n";
}
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The formular used here works like this:

1. if the month is January or february (1 or 2) shift it to the end of the preceeding year (month 13 and 14)
2. add 1 to the month
3. sum up the integer result of:
   1. month * 30.6
      30.6 gives it the correct toggle between 30 and 31. February is the last month, so no problem with 28/29
   2. year * 365.25
      this takes into account the leap years
   3. year / -100
      this subtracts the non-leap years
   4. year / 400
      this re-adds the non-non-leap years
4. the day should then be added

the reminder of 7 is the weekday starting with Sat==0