Find a Four-Digit Number Divisible by 2, 3, and 9Date: 11/29/2003 at 14:13:54 From: Bri Subject: four digit number divisible by 2,3,9 What is a four digit number that is divisible by 2, 3, and 9? For example, a three digit number that is divisible by 2 and 9 is 108. It can't be 1,000 because 3 can't go into it. It can't be 2,000 because 3 and 9 can't go into it. Date: 12/01/2003 at 14:10:02 From: Doctor Jason Subject: Re: four digit number divisible by 2,3,&amp;9 Hi Bri, Thanks for writing to Dr. Math! If a number is divisible by 9, then it is also divisible by 3, since 3 is a factor of 9 (3 * 3 = 9). If a number ends in 0, 2, 4, 6, or 8, then it is divisible by 2. So, what you need is an even, four-digit number that is divisible by 9. (Hint: If the sum of all of the digits of a number is divisible by 9, then the number itself is divisible by 9.) The first that comes to my mind is 9000. 9000 / 2 = 4500 9000 / 3 = 3000 9000 / 9 = 1000 To find more examples, you might just grab a calculator and enter even, four-digit numbers and see if they are divisible by 9. You could also just use all of the rules that we have listed above to "create" a number that fits the 3 criteria. 1. Start with four blank spaces representing the 4 digits in the number: ___ ___ ___ ___ 2. Fill in the ones digit with an even number, which takes care of the 'must be divisible by 2' rule: ___ ___ ___ _4_ 3. Next, fill in 2 of the remaining blanks with any numbers you want: _2_ _7_ ___ _4_ 4. Now add the 3 digits together. Subtract your sum from either 9, 18, 27, whichever is the next multiple of 9 that is greater than your sum. The difference is the number that will fill the last blank. So far, I have used 2, 7, and 4. The sum of those three digits is 13. I need to subtract 13 from 18, since 18 is the next multiple of 9 that is greater than 13: 18 - 13 = 5. The number 5 fills my last blank: _2_ _7_ _5_ _4_ 5. Check your answer: 2754 / 2 = 1377 2754 / 3 = 918 2754 / 9 = 306 Note that as long as we keep an even number in the ones position, we can rearrange the digits for even more possibilities since the sum of the digits will still be divisible by 9: 2574 / 9 = 286 5274 / 9 = 586 5724 / 9 = 636 4572 / 9 = 508 4752 / 9 = 528 7542 / 9 = 838 7524 / 9 = 836 7254 / 9 = 806 7425 / 9 = 825 Try it for yourself and let me know how it works out for you! - Doctor Jason, The Math Forum http://mathforum.org/dr.math/ |
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