Sugar - cuboid

Pablo received from his master a cuboid composed of identical sugar cubes, each weighing between 1000 and 2000. The Paul eat sugar cubes in layers.
On the first day, eat one layer from the front. On the second day, one layer is from the right, and on the third day, one layer is above. However, those three layers were each the same number of cubes. Find out how many cubes had the original cuboid. Find all possibilities.

Final Answer:

V₁ =  1320
V₂ =  1716

Step-by-step explanation:

$$\begin{gathered} V_1=1320 \\ 1320=11\cdot 10\cdot 12 \\ 11\cdot 10=10\cdot (12-1)=(11-1)\cdot (12-1)=110 \\ 1716=12\cdot 11\cdot 13 \\ {V}_1=12\cdot 11=11\cdot (13-1)=(12-1)\cdot (13-1)=132=1320 \end{gathered}$$

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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