minimum number

$\text{Unit digit is 7. After operation,\{ that is multiplying by 5 and} \\ \text {shifting 7 to fist position,\} 7 * 5=35, so unit digit must be 5.} \\ \text{So the last two digits would be 57.} \\ \color{#D61F06}{\text{Go to logical solution.} }\\$ ~ $\\ \text{ 57 * 5=285. So let us take the number as 2857.}\\ \text{7285/5=1457 and not 2857. So next step.}\\ \text{2857 * 5=14285 . So let us take the number as 142857.} \\ \text{142857 * 5=714285 so this is the number. } \\$

$\text{I do not know how this works. But this is a special number!!!}\\142857*1, ~142857*2,~142857*3,~142857*4~142857*5~142857*6~\\\text{all have the same digits shifted!!! But 142857 * 7=999999!!!}\\~$ $\color{#D61F06}{\text{Logical solution.}}\\~\text{Well the logical way is as follows.}\\\text{ x, y, z are unknown digits, xyz means 100+10y+z. }\\ \text{After the operation, the first digit is 7. So before the operation,} \\ \text{first two digits must be some thing more that 14. 14x * 5=714.} \\$ $\text{So x is only 2 so that we can have }7 \color{#D61F06}{1} 4. \\ \text {But 4 would come if next digit is 8, (8*5=40).} \\ \text {So the number likely is 142857. Check:- 142857*5 is equal to 714285.}$