Veamos.
Supongamos que tengamos las cifras a , b y c tales que a>b>c, donde el numeral mayor será<img src="https://tex.z-dn.net/?f=%5Coverline%7Babc%7D" /> y el menor será<img src="https://tex.z-dn.net/?f=%5Coverline%7Bcba%7D" /> .
Ahora restemos
<img src="https://tex.z-dn.net/?f=%5Coverline%7Babc%7D-%5Coverline%7Bcba%7D%3D%28100a%2B10b%2Bc%29-%28100c%2B10b%2Ba%29%5C%5C%20%5C%5C%0A%5Coverline%7Babc%7D-%5Coverline%7Bcba%7D%3D99%28a-c%29" />
Luego lo vemos de esta forma
<img src="https://tex.z-dn.net/?f=99%28a-c%29%3D11%5Ctimes%209%28a-c%29%5C%5C%20%5C%5C%0A%5Ctext%7BSi%20%7D9%28a-c%29%5Ctext%7B%20es%20de%20una%20cifra%2C%20entonces%20este%20es%20%7D%2099%5C%5C%20%5C%5C%0A%5Ctext%7BSi%20%7D9%28a-c%29%5Ctext%7B%20es%20de%20dos%20cifras%20o%20sea%20%7D%5Coverline%7Bxy%7D%5Ctext%7B%20entonces%20es%20%7D%5C%5C%20%5C%5C%0A11%5Ctimes%5Coverline%7Bxy%7D%3D%5Coverline%7B%5Cleft%28x%2B%5Cleft%5B%5Cdfrac%7Bx%2By%7D%7B10%7D%5Cright%5D%5Cright%29%5Cleft%28x%2By-10%5Cleft%5B%5Cdfrac%7Bx%2By%7D%7B10%7D%5Cright%5D%5Cright%29y%7D%5C%5C%20%5C%5C%0A%5Ctext%7Bdonde%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Cleft%5B%5Cdfrac%7Bx%2By%7D%7B10%7D%5Cright%5D%3D%5Cbegin%7Bcases%7D%0A0%26%5Ctext%7Bsi%20%7Dx%2By%5C%20%5Ctextless%20%5C%20%2010%5C%5C%0A1%26%5Ctext%7Bsi%20%7Dx%2By%5Cgeq%2010%0A%5Cend%7Bcases%7D%0A" />
<img src="https://tex.z-dn.net/?f=%5Ctext%7Bo%20sea%7D%5C%5C%20%5C%5C%0A11%5Ctimes%5Coverline%7Bxy%7D%3D%5Cbegin%7Bcases%7D%0A%5Coverline%7Bx%28x%2By%29y%7D%26%5Ctext%7B%2C%20si%20%7Dx%2By%3D9%5C%5C%0A%5Coverline%7B%28x%2B1%29%28x%2By-10%29y%7D%26%5Ctext%7B%2C%20si%20%7Dx%2By%5C%20%5Ctextgreater%20%5C%209%5C%5C%0A%5Cend%7Bcases%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctext%7Bsuma%20de%20cifras%20de%20%7D11%5Ctimes%5Coverline%7Bxy%7D%5C%5C%20%5C%5C%0A%5Csum%3D%5Cbegin%7Bcases%7D%0A2%28x%2By%29%26%5Ctext%7B%2C%20si%20%7Dx%2By%3D9%5C%5C%0A2%28x%2By%29-9%26%5Ctext%7B%2C%20si%20%7Dx%2By%5C%20%5Ctextgreater%20%5C%209%0A%5Cend%7Bcases%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctext%7Bcomo%20%7D%5Coverline%7Bxy%7D%5Cin%5C%7B18%2C27%2C36%2C...%2C%2081%5C%7D%5CLongrightarrow%20x%2By%3D9%5C%5C%20%5C%5C%0A%5Ctext%7BPor%20lo%20tanto%20la%20suma%20de%20cifras%20siempre%20es%20%7D2%28x%2By%29%3D2%289%29%3D18" />.