Sugar - cuboid

Pejko received from his master cuboid composed of identical sugar cubes with count between 1000 and 2000. The Pejko eat sugar cubes in layers.
The first day eat one layer from the front, second day one layer from right, the third day one layer above. Yet in those three layers were each the same number of cubes. Find out how many cubes had original cuboid. Determine all possibilities.


Result

V1 =  1320
V2 =  1716

Solution:

1320=111012  1110=10(121)=(111)(121)=110  1716=121113  1211=11(131)=(121)(131)=1321320 = 11 \cdot 10 \cdot 12 \ \\ \ \\ 11 \cdot 10 = 10 \cdot (12-1) = (11-1)\cdot (12-1) = 110 \ \\ \ \\ 1716 = 12 \cdot 11 \cdot 13 \ \\ \ \\ 12\cdot 11 = 11\cdot (13-1) = (12-1) \cdot (13-1) = 132
PHP program:

for($a=1; $a<1000; $a++)
{
    for($b=1; $b<1000; $b++)
    {
        for($c=1; $c<1000; $c++)
        {
            $V = $a*$b*$c;
            if($V > 2000 || $V<1000)
            {
                continue;
            }

            if($a*$b == ($c-1)*$b && ($c-1)*$b == ($a-1)*($c-1))
            {
                echo "$V = $a * $b * $c <br>";
            }
        }
    }
}




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