document.write( "Question 1066754: find the smallest number which is grater than 111,111,000 and divisible by 8 and 9 \n" ); document.write( "

Algebra.Com's Answer #682003 by KMST(5328) $\"\"$ $\"About$

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THE IDEA:
\n" ); document.write( "To be divisible by the relatively prime numbers $\"8\"$ and $\"9\"$ ,
\n" ); document.write( "a number has to be divisible by $\"8%2A9=72\"$
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\n" ); document.write( "THE HARD WAY:
\n" ); document.write( "Dividing $\"%22111%2C111%2C000%22\"$ by 72,
\n" ); document.write( "we get $\"%221%2C543%2C208%22\"$ as the quotient,
\n" ); document.write( "with $\"24\"$ as a remainder,
\n" ); document.write( "so $\"%221%2C543%2C208%22%2A72\"$ is less than $\"%22111%2C111%2C000%22\"$ ,
\n" ); document.write( "but the next multiple of $\"72\"$ is
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.
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\n" ); document.write( "EASIER:
\n" ); document.write( "Dividing $\"%22111%2C111%2C000%22\"$ by 72,
\n" ); document.write( "we get $\"%221%2C543%2C208%22\"$ as the quotient,
\n" ); document.write( "with $\"24\"$ as a remainder,
\n" ); document.write( "so adding $\"72-24=48\"$ to $\"%22111%2C111%2C000%22\"$
\n" ); document.write( "we get the next multiple of $\"72\"$ :
\n" ); document.write( " $\"%22111%2C111%2C000%22%2B48=highlight%28%22111%2C111%2C048%22%29\"$ .
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\n" ); document.write( "WITHOUT DIVIDING:
\n" ); document.write( "Like all numbers ending in $\"%22000%22\"$ ,
\n" ); document.write( " $\"%22111%2C111%2C000%22\"$ is divisible by $\"8\"$ ,
\n" ); document.write( "because $\"1000\"$ is divisible by $\"8\"$ .
\n" ); document.write( "When dividing by $\"9\"$ ,
\n" ); document.write( "the remainder can be found by adding the digits,
\n" ); document.write( "and repeating the digit adding with the result,
\n" ); document.write( "as many times as needed until you get a single digit result.
\n" ); document.write( "The final result is the remainder, unless it is $\"9\"$ .
\n" ); document.write( "If the final result is $\"9\"$ ,
\n" ); document.write( "then the number is divisible by $\"9\"$ ,
\n" ); document.write( "and the remainder is $\"0\"$ .
\n" ); document.write( "The sum of the digits of $\"%22111%2C111%2C000%22\"$ is $\"6\"$ ,
\n" ); document.write( "so when dividing $\"%22111%2C111%2C000%22\"$ by $\"9\"$ ,
\n" ); document.write( "we get a quotient $\"N\"$ , and the remainder is $\"6\"$ .
\n" ); document.write( "So, $\"%22111%2C111%2C000%22=9%2AN%2B6=3%2A3%2AN%2B3%2A2=3%283n%2B2%29\"$ is a multiple of $\"8\"$ ,
\n" ); document.write( "and it is even a multiple of $\"3\"$ ,
\n" ); document.write( "but it is not a multiple of $\"9\"$ .
\n" ); document.write( "If I add a multiple of $\"8\"$ that is also a multiple of $\"3\"$ ,
\n" ); document.write( "such as $\"3%2A8=24\"$ or $\"2%2A3%2A8=48\"$ ,
\n" ); document.write( "the sum will also be a multiple of $\"8\"$ and a multiple of $\"3\"$ .
\n" ); document.write( "One of those sums must be a multiple of $\"9\"$ too.
\n" ); document.write( " $\"%22111%2C111%2C000%22%2B24=%22111%2C111%2C024%22\"$ is not a multiple of $\"9\"$ ,
\n" ); document.write( "because adding digits we get
\n" ); document.write( " $\"%281%2B1%2B1%2B1%2B1%2B1%29%2B2%2B4=6%2B2%2B4=12\"$ and $\"1%2B2=3\"$ ,
\n" ); document.write( "but $\"%22111%2C111%2C000%22%2B48=highlight%28%22111%2C111%2C048%22%29\"$ is a multiple of $\"9\"$ ,
\n" ); document.write( "because $\"%281%2B1%2B1%2B1%2B1%2B1%29%2B4%2B8=6%2B4%2B8=18\"$ and $\"1%2B8=9\"$ . \n" ); document.write( "

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