The last two digits of powers of 7 cycle between 01, 07, 49, and 43.
The third digit is multiplied by 7 for 01 and 07, and is multiplied by 7 plus 3 for 49 and 43. Since you start off with the third digit being 0 (
![](https://latex.codecogs.com/png.latex?\bg_white 7^0 = 001)
) you get:
And this pattern repeats. The third digit goes in a cycle of 20 digits:
0 0 0 3 4 8 6 5 8 6 2 7 2 4 8 9 6 2 4 1
Now it starts at
![](https://latex.codecogs.com/png.latex?\bg_white 7^0)
So we want the 10,000th digit in this cycle. Which is the last digit since there are 20 digits in the cycle, which divides 10,000.
Hence the last 3 digits are 143.