$2100,4200,6300,8400$

hatchguy wrote: Find all positive integers that are equal to $700$ times the sum of its digits. Let number of digits be $k$, it's easy too see $3\leq k<5$ ,So checking cases we get it's also easy to see first digit is 2 times than neighbor digit,so all are $2100.4200,6300,8400$

700 time of digit is just the same as 7 times the sum of digits. If $ d $ is the number of digits, then the maximum value of the sum of digits is $ 63d $ and the minimum value is $ 10^{d-1} $. Therefore, the minimum value of $ d $ is 3. Let $ n=100a+10b+c $, where $ a,b,c $ are digits, then $100a+10b+c=7a+7b+7c $ and hence $ 93a+3b=6c $. Therefore $ a=0 $ and $ b=2c $. The only posibilities for b and c are $ (2,1), (4,2), (6,3) $, and $ (8,4) $, thus the only numbers that satisfy the problem are $ m=2100,4200,6300 \ \text{and} \ 8400 $.

Define by $s(n) =$ sum of digits of positive integer $n$, take m a positive integer such that $m = 700.s(m)$, it's easy to see that $m \equiv s(m) \pmod 9 \implies 700.s(m) \equiv s(m) \pmod 9\implies 9|699.s(m) \implies 3|s(m),$ so $3|m$, but $s(m)$ and $m$ are positive integers, then, $700.s(m) = m \implies 700|m,$ therefore $2100|m \implies m = 2100.k = 21.k.100 = 21k$ followed by 2 zeros at the end $\implies s(m) = s(21k) \implies 700.s(21k) = 2100k \implies s(21k) = 3k,$ but there is a property who says that: $s(a.b)$ is less than or equal to $s(a).s(b)$, using $a = 21, b = k,$ we get $s(21k)$ is less than or equal to $3.s(k) \implies s(21k) = 3k$ is less than or equal to $3.s(k)$, thus $k$ is less than or equal to $s(k)$, so, suppose that $k$ has $j$ digits, then $k > 10^j - 1$ and $s(k) < 9.j+1 \implies 10^j - 1$ is less than or equal to $9j+1$, therefore, $j = 1$ (only use an induction at $j$ to prove that inequality doesn't hold when $j > 1$) $\implies k < 10$, so $s(k) = k$. Testing the 10 cases we get only: $m = 2100, 4200, 6300, 8400.$

Obviously, It is at least a 3 digit number. 700 is not a solution. It cannot be 6 digit because it is too big. The largest sum of a 5 digit number(where the number is a multiple of 700)is 22. 22*700=15400. So, it is at least four digits but smaller than 15400. By trying(only 2*700 - 21*700), we can find clues and easily, we get the answer 2100, 4200, 6300 and 8400.