document.write( "Question 1140645: The 86-digit number 666...666 ( made entirely of sixes) is multiplied by the 85 digit-number 333...333 (made entirely of threes). How many times will the digit 7 appear in the product? \n" ); document.write( "

Algebra.Com's Answer #761174 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "85 times.

\n" ); document.write( "Look at the pattern:

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document.write( "66*33 = 2178    2 digits times 2 digits --> 1 7\r\n" );
document.write( "666*333 = 221778    3 digits times 3 digits --> 2 7's\r\n" );
document.write( "6666*3333 = 22217778    4 digits times 4 digits --> 3 7's\r\n" );
document.write( "66666*33333 = 2222177778    5 digits times 5 digits --> 4 7's\r\n" );
document.write( "...\r\n" );
document.write( "86 digits time 86 digits --> 85 7's

\n" ); document.write( "Here is a demonstration of the result for the case where both numbers are 5 digits; the demonstration for 86-digit numbers will be similar.

\n" ); document.write( " $\"66666%2A33333\"$
\n" ); document.write( "= $\"66666%2A%2899999%2F3%29\"$
\n" ); document.write( "= $\"66666%2A%28%28100000-1%29%2F3%29\"$
\n" ); document.write( "= $\"%286666600000-66666%29%2F3\"$
\n" ); document.write( "= $\"2222200000-22222\"$
\n" ); document.write( "= $\"2222177778\"$ \n" ); document.write( "

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