Find all the numbers that are equal to one quarter of the sum of their own digits

Question

The number $1.5$ is special because it is equal to one quarter of the sum of its digits, as $1+5=6$ and $\frac{6}{4}=1.5$ .Find all the numbers that are equal to one quarter of the sum of their own digits.

I was puzzling over this question for a while, but I couldn't find a formula without using the $\sum$ , but I can't really solve generalizations, only come up with them. The only thing I could come up with was to brute-force it, but I can't really come up with any 'special' numbers. Any help?

Answer 1

This is not much better than a brute force method:

• The sum of the digits $S$ is a non-negative integer, so a quarter of it $\frac{S}4$ is non-negative, of the form $x$ or $x.25$ or $x.5$ or $x.75$ for some non-negative integer $x$.

• We must have $x < 10$ since if $10 \le x\lt 100$ then $40 \le 4x \le S \lt 412$ so the sum of the digits of $x$ must be at least $40-12=28$ but two-digit integers have sums of digits no more than $18$. Similarly with larger $x$.

  • So a satisfactory $x$ has sum of digits $4x=S=x$ with solution $x=0$
  • and a satisfactory $x.25$ has sum of digits $4(x+\frac14)=S=x+7$ with solution $x=2$
  • and a satisfactory $x.5$ has sum of digits $4(x+\frac12)=S=x+5$ with solution $x=1$
  • and a satisfactory $x.75$ has sum of digits $4(x+\frac34)=S=x+12$ with solution $x=3$

making the numbers which work $$0, \quad2.25,\quad 1.5,\quad 3.75$$