Modular Arithmetic: Problem with the calculation

Question

I am trying to solve the following:

99^7 (mod 123) = 54
Applying formula:
=(99 * 99^2 * 99^4) (mod 123)
= [(99 mod 123) * (99 ^2 mod 123)  * (99^4 mod 123)] mod 123
= [24* 75 * 75 * 75] mod 123 = 10,125,000 mod 123 = 82,317 * 123 
=10,124,991=9

With out using the formula, I am getting 54 but if I use formula, then I get 9.

Somebody please guide me.

Zulfi.

Answer 1

Your working is incorrect. The answer is indeed 54.

Hint: What is $99^2 \pmod{123}$?