To find the last two digits of the number  3^123 + 7^123 + 9^123,  it is enough to know the last two digits of each number

3^123, 7*123 and 9^123, separately.



The last two digits of the numbers  3^1, 3^2, 3^3, . . . , 3^k, . . . repeat periodically with the period of 20.

    So (and in particular), two last digits of the numbers  3^21, 3^41, . . . , 3^121 are the same: they are 03.

    The two last digits of the number 3^123, therefore, is not difficult to calculate : they are 27.



The last two digits of the numbers  7^1, 7^2, 7^3, . . . , 7^k, . . . repeat periodically with the period of 4.

    So (and in particular), two last digits of the numbers  7^5, 7^9, . . . , 7^121 are the same: they are 07.

    The two last digits of the number 7^123, therefore, is not difficult to calculate : they are 43.



The last two digits of the numbers  9^1, 9^2, 9^3, . . . , 9^k, . . . repeat periodically with the period of 10.

    So (and in particular), two last digits of the numbers  9^11, 9^21, . . . , 9^121 are the same: they are 09.

    The two last digits of the number 9^123, therefore, is not difficult to calculate : they are 29.



Therefore, the last two digits of the number  3^123 + 7^123 + 9^123  you can easily find by taking the sum 

    27 + 43 + 29 = 99.


ANSWER.  The last two digits of the number  3^123 + 7^123 + 9^123  are  99.

    The fact that the last two digits of the sequence  3, 3^2, 3^3, . . . , 3^k . . . form a periodic sequence seems to be a miracle.

    But this fact is INEVITABLE consequence of simple properties.


    If to consider the last two digits, there are only finite number of their combinations: 

    00, 01, 02, . . . 10, 11, 12, . . . , 98, 99 --- in all, there are only 100 such 2-digit combinations.


    From the other side, the sequence  3, 3^2, 3^3, . . . , 3^k . . . is INFINITE.


    Therefore, by projecting it into the last two digits sequence, we INEVITABLY will have a repetition;

    and as soon as such a repetition will happen for the first time, the periodic behavior and the period itself are just provided.


    So, there is no any miracle in it - it is an inevitable fact.


    And this fact plays a KEY ROLE in similar proofs, making a key to solving such problems.


    Those who study Math from Math schools and/or from Math circles (or from associated Math books), usually know this remarkable property.


    Now, after reading my solution, you know this property, too (!)


    So from now, you do belong to a category of those lucky persons who know it (!)


    My congratulations (!) (!)