Sum of Digits of Multiples of NineDate: 08/12/2004 at 05:19:50 From: Saba Subject: number theory: multiples of 9 Why is it that when you add the individual digits of any multiple of nine until a single digit answer is reached the answer is always nine? Is it possible to prove this? For example, 99 => 9 + 9 = 18 => 1 + 8 = 9 Why doesn't it work with other numbers between 1-9 either? Date: 08/12/2004 at 10:10:23 From: Doctor Luis Subject: Re: number theory: multiples of 9 Hi Saba, Good job finding that pattern! The reason is that the sum of the digits of ANY multiple of 9 is also a multiple of 9. Since you keep adding the digits (each time getting a new multiple of 9, but a smaller multiple), eventually you'll end up with a single digit number. Eventually you'll get to the multiple 9 itself. Now, how do I know that the sum of the digits is always a multiple of 9? Suppose that a number N has digits a,b,c,d,...(from right to left), N = a + 10b + 100c + 1000d + ... = a + (b + 9b) + (c + 99c) + (d + 999d) + ... = (a + b + c + d + ...) + (9b + 99c + 999d + ...) = (a + b + c + d + ...) + 9*(b + 11c + 111d + ...) N = (sum of digits of N) + 9 * (some number) Now, look at that equation carefully. It means that (sum of digits of N) = N - 9 * (some number) Since N is assumed to be a multiple of 9, we can write it in terms of another integer k, so that N = 9k (sum of digits of N) = 9 * k - 9 * (some number) = 9 * (k - (some number)) = 9 * (some other number) Since we showed that the sum of the digits is 9 times some integer, then it is also a multiple of 9 itself. To summarize, starting from a multiple of 9, you keep adding the digits, each time arriving to a multiple of 9. This establishes a chain of decreasing multiples of 9, until eventually you reach 9 (from a two-digit multiple). Does that make sense? It doesn't work for other integers because the chain is broken. For example, multiples of 8 such as 56 don't add up to a multiple of 8. Well, I hope this helped! Let us know if you have any more questions. - Doctor Luis, The Math Forum http://mathforum.org/dr.math/ Date: 08/12/2004 at 12:48:19 From: Saba Subject: number theory: multiples of 9 Thanks loads for answering--your proof is very cool. You guys are the best! |
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