Algebra.Com's Answer #832813 by ikleyn(52817)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
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document.write( "The number 4^1000-1 is divisible by\r
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document.write( "a) 4
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document.write( "b) 5
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document.write( "c) 7
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document.write( "d) 13
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document.write( "e) 19
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document.write( " I just solved this problem and answered this question several days ago (~ one week ago) under this link\r
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document.write( "https://www.algebra.com/algebra/homework/equations/Equations.faq.question.1198925.html\r
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document.write( "https://www.algebra.com/algebra/homework/equations/Equations.faq.question.1198925.html\r
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document.write( " For your convenience, I copy-paste that my solution here again:\r
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document.write( "(a) Regarding (a), it is clear that the remainder of division  by 4 is -1, \r\n" );
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document.write( " which is the same as 3 mod(4); so, is not divisible by 4.\r\n" );
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document.write( "(b) Regarding (b), =  is divisible by  = 15;\r\n" );
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document.write( " so, is divisible by 5.\r\n" );
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document.write( "(c, d, e) Regarding (c), (d) and (e), I prepared a table below, which shows the remainders\r\n" );
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document.write( " of division  by 7, 13 and 19 respectively.\r\n" );
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document.write( " k mod(7) mod(13) mod(13)\r\n" );
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document.write( " 1 4 4 4\r\n" );
document.write( " 2 2 3 16\r\n" );
document.write( " 3 1 12 7\r\n" );
document.write( " 4 4 9 9\r\n" );
document.write( " 5 2 10 17\r\n" );
document.write( " 6 1 1 11\r\n" );
document.write( " 7 4 4 6\r\n" );
document.write( " 8 2 3 5\r\n" );
document.write( " 9 1 12 1\r\n" );
document.write( " 10 4 9 4\r\n" );
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document.write( " 3 6 9 <<<---=== the length of the period.\r\n" );
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document.write( " The table shows that for each of these divisors, 7, 13, 19, the sequence of remainders \r\n" );
document.write( " is periodical, as it should be.\r\n" );
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document.write( " Each period (in each column) starts from the very first term of the sequence.\r\n" );
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document.write( " For mod(7), the remainder 1 is every 3rd term of the sequence.\r\n" );
document.write( " Since the index/(the degree) of 1000 is not a multiple of 3,  mod(7) is not 1;\r\n" );
document.write( " hence, is not divisible by 7.\r\n" );
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document.write( " For mod(13), the remainder 1 is every 6th term of the sequence.\r\n" );
document.write( " Since the index/(the degree) of 1000 is not a multiple of 6, mod(13) is not 1;\r\n" );
document.write( " hence, is not divisible by 13.\r\n" );
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document.write( " For mod(19), the remainder 1 is every 9th term of the sequence.\r\n" );
document.write( " Since the index/(the degree) of 1000 is not a multiple of 9, mod(19) is not 1;\r\n" );
document.write( " hence, is not divisible by 19.\r\n" );
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document.write( "ANSWER. (a) is not divisible.\r\n" );
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document.write( " (b) is divisible.\r\n" );
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document.write( " (c) is not divisible.\r\n" );
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document.write( " (d) is not divisible.\r\n" );
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document.write( " (e) is not divisible.\r\n" );
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document.write( "Solved.\r
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