document.write( "Question 1127112: If 5^678 is written as an integer, what are it's final four digits?\r
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Algebra.Com's Answer #743484 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "The table below represents the values of   in the second column,  n = 1, 2, 3, 4, . . . , 20.\r\n" );
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document.write( "The third column represents  the last 4 digits of the number  .\r\n" );
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document.write( "1	5	           5\r\n" );
document.write( "2	25	          25\r\n" );
document.write( "3	125	         125\r\n" );
document.write( "4	625	        0625\r\n" );
document.write( "5	3125	        3125\r\n" );
document.write( "6	15625	        5625\r\n" );
document.write( "7	78125	        8125\r\n" );
document.write( "8	390625	        0625\r\n" );
document.write( "9	1953125	        3125\r\n" );
document.write( "10	9765625	        5625\r\n" );
document.write( "11	48828125	8125\r\n" );
document.write( "12	244140625	0625\r\n" );
document.write( "13	1220703125	3125\r\n" );
document.write( "14	6103515625	5625\r\n" );
document.write( "15	30517578125	8125\r\n" );
document.write( "16	152587890625	0625\r\n" );
document.write( "17	762939453125	3125\r\n" );
document.write( "18	3814697265625	5625\r\n" );
document.write( "19	19073486328125	8125\r\n" );
document.write( "20	95367431640625	0625\r\n" );
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document.write( "An experimental fact is that starting from n = 4, the last 4 digit numbers form a periodical (cyclical) sequence \r\n" );
document.write( "with the period (the cycle length) of 4:\r\n" );
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document.write( "    0625,  3125,  5625,  8125.\r\n" );
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document.write( "It can be proved mathematically.  I will not prove it here - see, for example, the lesson  \r\n" );
document.write( "    Find the last three digits of these numbers  in this site.\r\n" );
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document.write( "  is 678-th term of this long sequence, and it is equal to the  (678-3) mod 4  term of the basic cycle.\r\n" );
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document.write( "Since  (678-3) mod 4 = 3,  it means that the 678-th term of the long sequence    is equal \r\n" );
document.write( "to the 3-rd term of the basic cycle, i.e.  5625.\r\n" );
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document.write( "Answer.    has the last four digits  5625.  \r\n" );
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