document.write( "Question 1165085: Find the last two digits of the number 3^123 + 7^123 + 9^123. \n" ); document.write( "
Algebra.Com's Answer #789547 by ikleyn(52788)\"\" \"About 
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\n" ); document.write( "Find the last two digits of the number 3^123 + 7^123 + 9^123
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document.write( "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\r\n" );
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document.write( "3^123, 7*123 and 9^123, separately.\r\n" );
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document.write( "The last two digits of the numbers  3^1, 3^2, 3^3, . . . , 3^k, . . . repeat periodically with the period of 20.\r\n" );
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document.write( "    So (and in particular), two last digits of the numbers  3^21, 3^41, . . . , 3^121 are the same: they are 03.\r\n" );
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document.write( "    The two last digits of the number 3^123, therefore, is not difficult to calculate : they are 27.\r\n" );
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document.write( "The last two digits of the numbers  7^1, 7^2, 7^3, . . . , 7^k, . . . repeat periodically with the period of 4.\r\n" );
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document.write( "    So (and in particular), two last digits of the numbers  7^5, 7^9, . . . , 7^121 are the same: they are 07.\r\n" );
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document.write( "    The two last digits of the number 7^123, therefore, is not difficult to calculate : they are 43.\r\n" );
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document.write( "The last two digits of the numbers  9^1, 9^2, 9^3, . . . , 9^k, . . . repeat periodically with the period of 10.\r\n" );
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document.write( "    So (and in particular), two last digits of the numbers  9^11, 9^21, . . . , 9^121 are the same: they are 09.\r\n" );
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document.write( "    The two last digits of the number 9^123, therefore, is not difficult to calculate : they are 29.\r\n" );
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document.write( "Therefore, the last two digits of the number  3^123 + 7^123 + 9^123  you can easily find by taking the sum \r\n" );
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document.write( "    27 + 43 + 29 = 99.\r\n" );
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document.write( "ANSWER.  The last two digits of the number  3^123 + 7^123 + 9^123  are  99.\r\n" );
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\n" ); document.write( "\n" ); document.write( "A post-solution note\r
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document.write( "    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.\r\n" );
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document.write( "    But this fact is INEVITABLE consequence of simple properties.\r\n" );
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document.write( "    If to consider the last two digits, there are only finite number of their combinations: \r\n" );
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document.write( "    00, 01, 02, . . . 10, 11, 12, . . . , 98, 99 --- in all, there are only 100 such 2-digit combinations.\r\n" );
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document.write( "    From the other side, the sequence  3, 3^2, 3^3, . . . , 3^k . . . is INFINITE.\r\n" );
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document.write( "    Therefore, by projecting it into the last two digits sequence, we INEVITABLY will have a repetition;\r\n" );
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document.write( "    and as soon as such a repetition will happen for the first time, the periodic behavior and the period itself are just provided.\r\n" );
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document.write( "    So, there is no any miracle in it - it is an inevitable fact.\r\n" );
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document.write( "    And this fact plays a KEY ROLE in similar proofs, making a key to solving such problems.\r\n" );
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document.write( "    Those who study Math from Math schools and/or from Math circles (or from associated Math books), usually know this remarkable property.\r\n" );
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document.write( "    Now, after reading my solution, you know this property, too (!)\r\n" );
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document.write( "    So from now, you do belong to a category of those lucky persons who know it (!)\r\n" );
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document.write( "    My congratulations (!) (!)\r\n" );
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\n" ); document.write( "\n" ); document.write( "Come again to this forum soon to learn something new (!)\r
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