Henry Ernest Dudeney/Puzzles and Curious Problems/83 - Three Different Digits/Solution/Proof 2

Puzzles and Curious Problems by Henry Ernest Dudeney: $83$

Three Different Digits

Find all $3$-digit numbers with distinct digits that are divisible by the square of the sum of those digits.

Solution

$162$, $243$, $324$, $392$, $405$, $512$, $605$, $648$, $810$ and $972$.

Proof

Here is the source code of a program in C which finds them all:

   #include <stdio.h>
   
   int main()
   {
       for (int i = 100; i < 1000; i++)
       {
           int a = i / 100;
           int b = (i / 10) % 10;
           int c = i % 10;
           if (a != b && b != c && c!= a && i % ((a + b + c)*(a + b + c)) == 0)
               printf("%d, ", i);
       }
   }

Output:

 162, 243, 324, 392, 405, 512, 605, 648, 810, 972, 

Try it online!

$\blacksquare$