Question 1208429
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Greenestamps has a great solution. I'll provide another route.


Use the process described on <a href="https://www.algebra.com/algebra/homework/divisibility/lessons/successive-squaring.lesson">this page</a> to evaluate the following
3^123 = 27 (mod 100)
7^123 = 43 (mod 100)
9^123 = 29 (mod 100)
Each item can be verified with technology. Some examples are WolframAlpha, GeoGebra, spreadsheet, etc.


The results we got were 27, 43, and 29. 
They are the last two digits of 3^123, 7^123, and 9^123 in that exact order.


Therefore, 27+43+29 = <font color=red>99</font> are the last two digits of 3^123 + 7^123 + 9^123.


Verification with WolframAlpha
<a href="https://www.wolframalpha.com/input?i=%283%5E123+%2B+7%5E123+%2B+9%5E123%29+mod+100">https://www.wolframalpha.com/input?i=%283%5E123+%2B+7%5E123+%2B+9%5E123%29+mod+100</a>
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