document.write( "Question 1141460: My child has been given this question in homework.\r
\n" ); document.write( "\n" ); document.write( "I am thinking of a 3 digit number
\n" ); document.write( "When it is divided by 9, the remainder is 3
\n" ); document.write( "When it is divided by 2, the remainder is 1
\n" ); document.write( "When it is divided by 5, the remainder is 4
\n" ); document.write( "What is my number? \n" ); document.write( "
Algebra.Com's Answer #762005 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
My child has been given this question in homework.\r
\n" ); document.write( "\n" ); document.write( "I am thinking of a 3 digit number
\n" ); document.write( "When it is divided by 9, the remainder is 3
\n" ); document.write( "When it is divided by 2, the remainder is 1
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document.write( "We are asked to find an integer that is all three of these.\r\n" );
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document.write( "1. 3 more than a multiple of 9\r\n" );
document.write( "2. 1 more than a multiple of 2, which is to say it's odd.\r\n" );
document.write( "3. 4 more than a multiple of 5.\r\n" );
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document.write( "If we add a multiple of 9, 2 or 5, respectively to a multiple of 9, 2 or 5,\r\n" );
document.write( "respectively, we'll get another multiple of 9, 2 or 5, respectively.  Any\r\n" );
document.write( "multiple of all three, 9, 2, and 5 is a multiple of 90.\r\n" );
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document.write( "Therefore if we can find any number that satisfies 1,2, and 3 above, we can add\r\n" );
document.write( "any multiple of 90 to it and get another solution.  This problem calls for a 3\r\n" );
document.write( "digit solution, but if we can find a 1 or 2 digit solution, we can add 90 to it\r\n" );
document.write( "one or more times and get a 3 digit number that will also satisfy 1, 2, and 3.\r\n" );
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document.write( "We look at property 1.  To get an odd number when adding 3, we must begin with\r\n" );
document.write( "an even number\r\n" );
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document.write( "So we'll list the even multiples of 9, which are 18, 36, 54, 72, 90...\r\n" );
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document.write( "We add 3 to each and get 21, 39, 57, 75, 93 …\r\n" );
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document.write( "We look at property 3, that it is 4 more than a multiple of 5.\r\n" );
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document.write( "Every multiple of 5 ends in 0 or 5.  4 more than a number ending in 0 ends\r\n" );
document.write( "with 4, which would make it even.  4 more than a number ending in 5 ends\r\n" );
document.write( "with 9, which would make it odd.  We want an odd number, so it must end\r\n" );
document.write( "in 9, and the only candidate listed above that ends with 9 is 39.  It's true\r\n" );
document.write( "that 39 satisfies all three properties 1,2, and 3. That's becayuse\r\n" );
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document.write( "39 = 4×9+3 = 2×19+1 = 5×7+4\r\n" );
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document.write( "Trouble is, 39 only has two digits, so it doesn't qualify as a solution, but we\r\n" );
document.write( "can add 90 to it any number of times we wish, as long as we get a 3-digit\r\n" );
document.write( "number.  So adding 90 just once to 39 gives the three-digit number 129, and\r\n" );
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document.write( "129 = 4×18+3 = 2×64+1 = 5×25+4\r\n" );
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document.write( "We can keep getting more solutions by adding 90 over and over. There are 10\r\n" );
document.write( "3-digit solutions to your problem:\r\n" );
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document.write( "129 = 9×14+3 = 2×64+1 = 5×25+4\r\n" );
document.write( "219 = 9×24+3 = 2×109+1 = 5×43+4\r\n" );
document.write( "309 = 9×34+3 = 2×154+1 = 5×61+4\r\n" );
document.write( "399 = 9×44+3 = 2×199+1 = 5×79+4\r\n" );
document.write( "489 = 9×54+3 = 2×244+1 = 5×97+4\r\n" );
document.write( "579 = 9×64+3 = 2×289+1 = 5×115+4\r\n" );
document.write( "669 = 9×74+3 = 2×334+1 = 5×133+4\r\n" );
document.write( "759 = 9×84+3 = 2×379+1 = 5×151+4\r\n" );
document.write( "849 = 9×94+3 = 2×424+1 = 5×169+4\r\n" );
document.write( "939 = 9×104+3 = 2×469+1 = 5×187+4\r\n" );
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document.write( "Edwin
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