Recently I stumbled across a problem that is as follows:
Is it possible to find a number K such that there is a three-digit number that is K times the sum of its digits (a K-up number)?
I started by writing an equation:
$100a+10b+c=k(a+b+c)$
However, I couldn't find any relation between a, b, and c, or what will happen to them as k changes.
According to the given solution, there are 180 three-digit numbers that follow this rule. However, they are separated into different sets of numbers when K= a different number each time. For example, when
$K=37$:
$111=37(1+1+1)$
$222=37(2+2+2)$
$333=37(3+3+3)$
$...$
$999=37(9+9+9)$
There are 9 three-digit numbers that qualify when K=37, and $a=b=c$ (the three digits of the number)
Therefore, when K=37,
$a=\frac{3b}{7}+\frac{4c}{7}$
$b=\frac{7a}{3}+\frac{4b}{3}$
$c=\frac{7a}{4}+\frac{3c}{4}$
It may be time consuming to find one of these number by just randomly coming up with a number for K and another one for either a, b, or c.
My question is:
- Is there a general rule for all values of K?
- Is there a specific relationship between the digits a, b, and c as K changes?
Not really an answer, but data.
Below is a complete list of three-digit "$k$-up" numbers (that is, numbers that when divided by the sum of their digits, yield $k$), sorted by $k$. If there's an overall pattern, I don't see it. (Note that the $37$-up numbers include more than the multiples of $111$.)
$$\begin{array}{c:l} k \\ \hline 11 & 198 \\ \hline 12 & 108 \\ \hline 13 & 117, 156, 195 \\ \hline 14 & 126 \\ \hline 15 & 135 \\ \hline 16 & 144, 192, 288 \\ \hline 17 & 153 \\ \hline 18 & 162 \\ \hline 19 & 114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 399 \\ \hline 20 & 180 \\ \hline 21 & 378 \\ \hline 22 & 132, 264, 396 \\ \hline 23 & 207 \\ \hline 24 & 216 \\ \hline 25 & 150, 225, 375 \\ \hline 26 & 234, 468 \\ \hline 27 & 243, 486 \\ \hline 28 & 112, 140, 224, 252, 280, 308, 336, 364, 392, 448, 476, 588 \\ \hline 29 & 261 \\ \hline 30 & 270 \\ \hline 31 & 372, 465, 558 \\ \hline 32 & 576 \\ \hline 33 & 594 \\ \hline 34 & 102, 204, 306, 408 \\ \hline 35 & 315 \\ \hline 36 & 324, 648 \\ \hline 37 & 111, 222, 333, 370, 407, 444, 481, 518, 555, 592, 629, 666, 777, 888, 999 \\ \hline 38 & 342, 684 \\ \hline 39 & 351 \\ \hline 40 & 120, 240, 360, 480 \\ \hline 41 & 738 \\ \hline 42 & 756 \\ \hline 43 & 516, 645, 774 \\ \hline 44 & 792 \\ \hline 45 & 405 \\ \hline 46 & 230, 322, 414, 460, 506, 552, 644, 690, 736, 782, 828, 874, 966 \\ \hline 47 & 423, 846 \\ \hline 48 & 432, 864 \\ \hline 49 & 441, 735, 882 \\ \hline 50 & 450 \\ \hline 51 & 918 \\ \hline 52 & 312, 624, 780, 936 \\ \hline 53 & 954 \\ \hline 54 & 972 \\ \hline 55 & 110, 220, 330, 440, 550, 605, 660, 715, 770, 825, 880, 935, 990 \\ \hline 56 & 504 \\ \hline 57 & 513 \\ \hline 58 & 522, 870 \\ \hline 59 & 531 \\ \hline 60 & 540 \\ \hline 61 & 732, 915 \\ \hline 64 & 320, 512, 640, 704, 832, 960 \\ \hline 67 & 201, 402, 603, 804 \\ \hline 68 & 612 \\ \hline 69 & 621 \\ \hline 70 & 210, 420, 630, 840 \\ \hline 73 & 511, 730, 803 \\ \hline 76 & 912 \\ \hline 78 & 702 \\ \hline 79 & 711 \\ \hline 80 & 720 \\ \hline 82 & 410, 820, 902 \\ \hline 85 & 510 \\ \hline 89 & 801 \\ \hline 90 & 810 \\ \hline 91 & 910 \\ \hline 100 & 100, 200, 300, 400, 500, 600, 700, 800, 900 \\ \hline \end{array}$$