Four Digit Number - Can You Find It?

First of all, let's set our number $n=1000a+100b+10c+d$ , where $a$ , $b$ , $c$ , and $d$ are distinct integers in the interval $[1,9]$ . Note that none of the integers can be $0$ because we can't make the first digit $0$ as the number would no longer have $4$ digits, which would result in less than $23$ possible permutations.

Let's assume that our number $n$ is the $24th$ permutation. In counting these permutations, we notice that each digit must be in each spot an equal number of times. There are $4$ variables $(a,b,c,d)$ and $24$ possible permutations, meaning that each variable must be in each spot $\frac{24}{4}=6$ times.

If the sum of the $23$ permutations is $157193$ , then $157193+n$ should be all $24$ . Taking into account that every variable should be in each place $6$ times:

$157193+n=6666(a+b+c+d) \Longrightarrow 157193+n \equiv 0$ (mod $6666$ )

$157193+n \equiv 3875+n \equiv 0$ (mod $6666$ ) $\Longrightarrow n \equiv 2791$ (mod $6666$ )

This means that $n=2791+6666m$ . There are two integer solutions that result in $4$ digit numbers: $2791$ and $9457$ . Let's plug them into our original equation to check which one works:

$157193+2791=159984\neq126654=6666(2+7+9+1)$

$157193+9457=166650=6666(9+4+5+7)$

Therefore our answer must be $\boxed{9457}$ .

Since all 24 permutations of $\overline{abcd}$ would have each digit appear 6 times in each position, they add up to 6666(a+b+c+d). So $\overline{abcd}$ bust be a multiple of 6666 minus 157193. The only candidates for $\overline{abcd}$ are :

$24×6666-157193=2791$ but $2+7+9+1 \neq 24$

$25×6666-157193=9457$ and $9+4+5+7=25$

10^3a + 10^2b + 10c + d

including the original number = 6666 * (a + b + c + d)

157193 + 10^3a + 10^2b + 10c + d = 6666 * (a + b + c + d)

Easiest way to proceed is by trial and error:

1. Locate the next multiple of 6666 that is greater than 157193.

2. Check to see if the number 159984 satisfies the equation, which it does not.

6666 * 25 = 166650

166650 - 157193= 9457

9 + 4 + 5 + 7 = 25

Therefore, the answer is the number 9457 .