document.write( "Question 996686: What is the sum of the digits of the product of 23 ones and 32 tens? \n" ); document.write( "

Algebra.Com's Answer #615020 by KMST(5345) $\"\"$ $\"About$

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MULTIPLYING:
\n" ); document.write( "You get $\"23%2A320=7360\"$ , and
\n" ); document.write( "adding the digits you get $\"7%2B3%2B6%2B0=highlight%2816%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "WITHOUT MULTIPLYING:
\n" ); document.write( " $\"23%2A320=23%2A32%2A10\"$ ,
\n" ); document.write( "so the product $\"23%2A320\"$ is $\"23%2A32\"$ tens,
\n" ); document.write( "with the same digits as $\"23%2A32\"$ with a $\"0\"$ added to the right.
\n" ); document.write( "We need to find the sum of the digits of $\"23%2A32\"$ .
\n" ); document.write( "If you add the digits of a number, and add the digits of the sum, and so on, until you get a 1-digit number, the final result is the remainder of dividing the original number by 9.
\n" ); document.write( "Since $\"2%2B3=5\"$ ,
\n" ); document.write( "when we divide $\"23\"$ by $\"9\"$ , the remainder is $\"5\"$ , and
\n" ); document.write( "when we divide $\"32\"$ by $\"9\"$ , the remainder is $\"5\"$ .
\n" ); document.write( "When you multiply numbers, the remainder from dividing the product by 9
\n" ); document.write( "is the the remainder of dividing the product of the remainders of dividing the factors by 9.
\n" ); document.write( "So, the remainder from dividing the product $\"23%2A32\"$ by 9 is
\n" ); document.write( "the remainder of dividing $\"5%2A5=25\"$ by 9.
\n" ); document.write( "which is $\"2%2B5=7\"$ .
\n" ); document.write( "However, $\"23%2A32\"$ is number ending in $\"2%2A3=6\"$ , and
\n" ); document.write( "Is a 3-digit number between $\"20%2A30=600\"$ and $\"25%2A32=800\"$ .
\n" ); document.write( "So, the sum of the first and last digits of that product is greater than $\"6%2B6=12\"$ , and
\n" ); document.write( "the sum of all 3 digits of $\"23%2A32\"$ must be less than $\"7%2B9%2B6=22\"$ .
\n" ); document.write( "Then, the sum of all 3 digits of $\"23%2A32\"$ must be a 2-digit number between $\"12\"$ and $\"22\"$ ,
\n" ); document.write( "The only such number that divided by 9 has a remainder of $\"7\"$ is
\n" ); document.write( " $\"9%2B7=highlight%2816%29\"$ ,
\n" ); document.write( "because $\"2%2A9%2B7=18%2B7=25\"$ .\r
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