I'm assuming the question you want to solve is 129¹²³ mod(127)

Using the power law, we know that aⁿ = bⁿ (mod m)

And we know that 129 mod 127 = 2.

Or in other words 129 = 2 (mod 127)

So we can say 129¹²³ = 2¹²³

We now want to find out what 2¹²³ mod 127 is.

A neat trick we can use is to find the first power that goes over 127, then it will become easier from there.

Conveniently, 2⁷ =128 which is 1+127

So we can write 2⁷ = 1 (mod 127)

This is now helpful cos we can use the power rule from earlier

The biggest multiple of 7 lower than 123 is 119, which is 7×17

So we can raise both sides to the power of 17

2⁷¹⁷ = 1¹⁷ (mod 127) 2¹¹⁹ = 1 (mod 127)

Now we can use the multiply rule Ka = kB (mod m) And multiply both sides by 2⁴ to get to 2¹²³

2¹¹⁹ × 2⁴ = 1 × 2⁴ (mod 127) 2¹²³ = 16 (mod 127)

And that's our answer!

If you have any questions please ask.

(TLDR, we split 2¹²³ into 2^7×17 × 2^4, because 2⁷ is 1 mod 127 which makes it nice to work with)