This is how I did it using pattern recognition.
First, observe the numbers in bold and the underlined:
101^1 = 101
101^2 = 10201
101^3 = 1030301
101^4 = 104060401
We can see that for each successive power of 101, the last two digits remain the same, but the third digit is the same as power. Therefore, for 100,
101^100 = something10001.
Therefore : 101^100 - 1 = something10000
Hence the last 5 digits are 10000
Let me know if that helps
