document.write( "Question 774935: what is the remainder when 5^2 + 5^3 + 5^4 + .......+ 5^247 is divided by 52 \n" ); document.write( "
Algebra.Com's Answer #472608 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "52+53+54+...+5247 =\r\n" );
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document.write( "(52+53+54+55)+54(52+53+54+55)+58(52+53+54+55)+512(52+53+54+55)+ ... +5240(52+53+54+55)+5244(52+53) =\r\n" );
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document.write( "(3900)+54(3900)+58(3900)+512(3900)+ ... +5240(3900)+5244(150)\r\n" );
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document.write( "Since 3900 is divisible by 52 we only need to look at the\r\n" );
document.write( "remainder when 5244(150) is divided by 52\r\n" );
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document.write( "5244(150) = (5244-1+1)(150) = (5244)(150)-1(150)+1(150) = 150[(5244)-1]+150=150[(54)61-1] + 150\r\n" );
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document.write( "Since xk-1 is divisible by x-1,\r\n" );
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document.write( "(54)61-1 is divisible by 54-1 or 624 which is divisible by 52, so\r\n" );
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document.write( "we only need to find the remainder when 150 is divided by 52\r\n" );
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document.write( "     2\r\n" );
document.write( "52)150\r\n" );
document.write( "   104\r\n" );
document.write( "    46\r\n" );
document.write( "Answer: 46\r\n" );
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document.write( "Edwin

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