Algebra.Com's Answer #831370 by ikleyn(52832)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
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document.write( "Find the last two digits in the decimal numeral for (3¹⁶⁷)⁹⁵
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document.write( "A) 27
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document.write( "B) 43
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document.write( "C) 81
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document.write( "D) 29
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document.write( "E) 07
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document.write( "(1) First, let's find two last digits of the number  .\r\n" );
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document.write( " It is clear, that looking for the last two digits of the number , \r\n" );
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document.write( " we should track last two digits of the number of  for k = 1, 2, 3, 4, . . . \r\n" );
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document.write( " This sequence of the last two digits of numbers {3^k mod 100} is periodical. \r\n" );
document.write( " The period starts from the first term and the period length is 20, \r\n" );
document.write( " as it can be seen from this table below\r\n" );
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document.write( " k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21\r\n" );
document.write( " 3^k mod 100 3 9 27 81 43 29 87 61 83 49 47 41 23 69 7 21 63 89 67 1 3\r\n" );
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document.write( " Since 167 = 8*20 + 7, the last two digits of the number is the 7th term of this \r\n" );
document.write( " cyclic sequence, i.e. the number 87.\r\n" );
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document.write( "(2) So, now next goal is to determine the last two digits of the number  .\r\n" );
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document.write( " Again, it is clear, that looking for the last two digits of the number , \r\n" );
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document.write( " we should track last two digits of the numbers {{87^k}}} for k = 1, 2, 3, 4, . . . \r\n" );
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document.write( " This sequence of the last two digits of numbers {87^k mod 100} is periodical.\r\n" );
document.write( " The period starts from the first term and the period length is 20, \r\n" );
document.write( " as it can be seen from this table below\r\n" );
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document.write( " k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21\r\n" );
document.write( " 87^k mod 100 87 69 3 61 7 9 83 21 27 49 63 81 47 89 43 21 67 29 23 1 87\r\n" );
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document.write( " Since 95 = 4*20 + 15, the last two digits of the number is the 15th term of this \r\n" );
document.write( " cyclic sequence, i.e. the number 43.\r\n" );
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document.write( "ANSWER. The last two digits in the decimal numeral for (3¹⁶⁷)⁹⁵ are 43.\r\n" );
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document.write( "Solved.\r
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document.write( "To see many other similar (and different) solved problems of this type, look into the lessons\r
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document.write( " - What is the last digit of the number a^n ? \r
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document.write( " - Find the last three digits of these numbers \r
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document.write( " - Find the last two digits of the number 3^123 + 7^123 + 9^123 \r
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document.write( "in this site.\r
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document.write( "////////////////\r
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document.write( "@MathLover1 copy-pasted her response from this web-page\r
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document.write( "https://www.quora.com/What-are-the-last-2-digits-of-the-decimal-numeral-for-3-167-95\r
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document.write( "but forgot to refer to the source.\r
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