Good catch! As originally coded, the error rate across all M/N combinations (< 2**32), seems to come out at ~ 1 in 15 (6.66%).

#! perl -slw
use strict;
use List::Util qw[ min ];
use Math::Random::MT qw[ rand ];

$|++;

sub check {
    my( $m, $n ) = @_;
    my $step = ( $m +1 ) / $n;
    my $f = int( $n * $step ) -1;
    return if  $f != $m;
    return 1;
}

my $trials = 0;
my $fails = 0;
for ( 1 .. 1e6 ) {
    my $m = int( rand 2**32 );
    for ( 1 .. min( $m, 1000 ) ) {
        ++$trials;
        my $n = 1+int( rand $m );
        check( $m, $n ) or ++$fails;
    }
    printf "\r$_ : %f%%", $fails *100 / $trials;
}
__END__
C:\test>ranges
76977645 : 6.620262%
[download]

However, a simple fudge floating point rounding correction factor of 0.000001 seems to sort things out nicely:

#! perl -slw
use strict;
use List::Util qw[ min ];
use Math::Random::MT qw[ rand ];

$|++;

sub check {
    my( $m, $n ) = @_;
    my $step = ( $m +1.000001 ) / $n;
    my $f = int( $n * $step ) -1;
#    warn( "m:$m n:$n f:$f\n" )
    return if  $f != $m;
    return 1;
}

my $trials = 0;
my $fails = 0;
for ( 1 .. 1e6 ) {
    my $m = int( rand 2**32 );
    for ( 1 .. min( $m, 1000 ) ) {
        ++$trials;
        my $n = 1+int( rand $m );
        check( $m, $n ) or ++$fails;
    }
    printf "\r$_ : %f%%", $fails *100 / $trials;
}


__END__
 C:\test>ranges
6783635 : 0.000000%
[download]