document.write( "Question 1095973: Using binomial theorem what is the Last three digits of the number 27^27? \n" ); document.write( "

Algebra.Com's Answer #710528 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "1.   = .\r\n" );
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document.write( "    Let us go up on this upstairs from the bottom to the top step by step.\r\n" );
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document.write( "2.  First consider the number  .  It is  19683:\r\n" );
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document.write( "     = 19683.\r\n" );
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document.write( "     Write it in the form   = .\r\n" );
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document.write( "     It is clear that for    and  for    the last three digits are determined by last \r\n" );
document.write( "     \r\n" );
document.write( "     three digits \"683\" of the number  .\r\n" );
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document.write( "     The part    does not affect the last three digits of the number  .\r\n" );
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document.write( "     It is exactly what the binomial theorem says and provides in this situation.\r\n" );
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document.write( "     Therefore, in finding the three last digits of the number    we can track only  for    and do not concern about other terms.\r\n" );
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document.write( "     It implies that the last three digits of the number    are exactly the same as the last three digits of the number  .\r\n" );
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document.write( "     The number   = 318611987,  as easy to calculate (I used Excel in my computer),  so its last three digits are  987.\r\n" );
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document.write( "3.  Now we can make the next (and the last) step up on this upstairs in the same way.\r\n" );
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document.write( "    The last three digits of the number    are the same as the last three digits of the number  ,  and it is easy to calculate.\r\n" );
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document.write( "     = 961504803,    and its last three digits are  803.\r\n" );
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document.write( "    Therefore, the last three digits of the number    are 803.\r\n" );
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