Sums of Consecutive Integers with Digital Sums
Date: 01/27/2004 at 00:20:54 From: Patrick Subject: 100 Find all sets of positive consecutive integers that meet the following two conditions: 1. The integers sum to one hundred. 2. The digits of the integers sum to a number greater than 30.
Date: 01/27/2004 at 16:25:16 From: Doctor Ian Subject: Re: 100 Hi Patrick, At first I was going to ask if this just referred to pairs of consecutive integers, but that can't be the case, because there are _no_ pairs like that. (The closest ones would be 49 + 50 = 99 and 50 + 51 = 101.) So now we have to consider any number of integers. Which makes this a pretty interesting problem! The hard thing is that there just seem to be so many possibilities. But is this really the case? We've already ruled out all the pairs, and all the integers greater than 48! Maybe we can rule out some more in the same way. For example, suppose we have three consecutive numbers. Then they're going to have to be around 33: 32 + 33 + 34 = 99 33 + 34 + 35 = 102 So there can't be any groups of three, and we don't have to worry about any numbers greater than 31. So we're making progress. Let's try groups of four. The numbers will have to be around 25: 23 + 24 + 25 + 26 = 98 24 + 25 + 26 + 27 = 102 You can keep going this way, but how long will that continue? How large a group would you have to consider? Note that to get close for a group of a particular size, we can figure out an average size by dividing 100 by our number: 2 times an average of 50 = 100 3 times an average of 33 = about 100 4 times an average of 25 = 100 5 times an average of 20 = 100 6 times an average of 18 = about 100 and so on. Then we're going to pick values around the average, with about half above, and the other half below. What if we try something like 20 times an average of 5 = 100 That looks good, but if we think about it, we see that it can't work, because we'd have to start considering negative numbers. (Do you see why? About half of those values are going to have to be less than 5.) So there is a limit to the size of the groups we have to consider, and it's less than 20. There's still some work to do, but can you take it from here? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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