document.write( "Question 800889: A student found that all three digit numbers of the form abc, where a+b is a multiple of 7, are divisible by 7. She would like to know why. Give an algebraic expression. \n" ); document.write( "
Algebra.Com's Answer #483361 by KMST(5328)\"\" \"About 
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That student was wrong, and a single counterexample can prove her wrong.
\n" ); document.write( "770 is a 3 digit number that is a multiple of 7, and its first two digits add up to 14, which is a multiple of 7.
\n" ); document.write( "However, 771, 772, 773, 774, 775, 776, 778, and 779, with the same two first digits are not multiples of 7.
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\n" ); document.write( "I propose a different divisibility by 7 rule.
\n" ); document.write( "All three digit numbers of the form aba, where a+b is a multiple of 7, are divisible by 7.
\n" ); document.write( "The value of a three digit numbers of the form aba is
\n" ); document.write( "\"100a%2B10b%2Ba\"
\n" ); document.write( "If \"a%2Bb\" is a multiple of 7, either \"a%2Bb=7\" or \"a%2Bb=14\".
\n" ); document.write( "Splitting the proof into two cases makes it easier to see and write.
\n" ); document.write( "(Otherwise I have to write that \"a%2Bb=7n\" where \"n\" is an integer).
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\n" ); document.write( "If \"a%2Bb=7\", then \"b=7-a\", and substituting we find the value of aba to be
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\n" ); document.write( "Since \"a\" is a digit, \"13a\" is an integer, and so is \"13a%2B10\".
\n" ); document.write( "Then \"aba=7%2A%2813a%2B10%29\" divides evenly by 7 and the quotient is \"13a%2B10\".
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\n" ); document.write( "If \"a%2Bb=14\", then \"b=14-a\", and substituting we find the value of aba to be
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\n" ); document.write( "Since \"a\" is a digit, \"13a\" is an integer, and so is \"13a%2B20\".
\n" ); document.write( "Then \"aba=7%2A%2813a%2B20%29\" divides evenly by 7 and the quotient is \"13a%2B20\". \n" ); document.write( "
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