Algebra.Com's Answer #808671 by ikleyn(52782)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Prove 4•10^(2n)+9•10^(2n-1)+5 is divisible by 99 for n is in N.
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document.write( "The given number is\r\n" );
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document.write( "  . (1)\r\n" );
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document.write( " | Notice that the number  gives the remainder 1, |\r\n" );
document.write( " | when is divided by 9, for ANY positive integer m. |\r\n" );
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document.write( "It implies that the first addend in (1) gives the remainder 4, when is divided by 9.\r\n" );
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document.write( "The second addend in (1) is divisible by 9 without the remainder (OBVIOUSLY).\r\n" );
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document.write( " It makes it OBVIOUS that the number (1) is divisible by 9 without the remainder.\r\n" );
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document.write( "Next, the number (1) has the form 49000 . . .005, where \r\n" );
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document.write( " - the leading digit 4 is located in some O D D position (2n+1), counting from the most right \"ones\" position;\r\n" );
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document.write( " - the digit 9 is placed in the next (E V E N) position (2n);\r\n" );
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document.write( " - and the last, \"ones\" digit 5 is located in the most right position number ONE (counting from the right);\r\n" );
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document.write( " - while all the other digits are zeros.\r\n" );
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document.write( "Applying the \"divisibility by 11 rule\", we see that the number (1) has alternate sum of digits 4 - 9 + 5 = 0.\r\n" );
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document.write( " Since the alternate sum of digits equal 0 (i.e. is divisible by 11),\r\n" );
document.write( " it means that the number (1) itself is divisible by 11,\r\n" );
document.write( " according to the \"divisibility by 11 rule.\r\n" );
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document.write( "Thus the number (1) is multiple of 9 and 11 --- HENCE, it is multiple of 99.\r\n" );
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document.write( "Solved (mentally), explained and completed.\r
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document.write( "On the \"divisibility by 11 rule\" see the lesson\r
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document.write( " - Divisibility by 11 rule \r
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document.write( " An integer number is divisible by 11 if and only if \r
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document.write( " the alternate sum of its digits is divisible by 11.\r
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