Sums of Square Integers PuzzleDate: 07/01/2002 at 20:38:24 From: Janette Subject: "problem of the week" problem I have this problem of the week that no one gets in school and I was wondering if you could help me. Here's the problem: How many numbers from 1-100 can be expressed as the sum of the squares of two positive integers? A girl in school told me that this was a college problem. Please help me!! Thank you, Janette M. Casarrubias Date: 07/02/2002 at 02:32:19 From: Doctor Ian Subject: Re: "problem of the week" problem Hi Janette, This is one of those problems that seems harder than it actually is. For one thing, it seems as though you'd have to check all the numbers from 1 to 100! And for each one, you'd have to try all the different ways that it could be the sum of two squares. That sounds like an awful lot of work. And if you approached it that way, it _would_ be a lot of work. But you can approach it from a different direction. Consider that 10^2 is 100, and 100 = 10^2 + 0^2 Since 0^2 isn't the square of a positive integer, we know that all of the integers being squared have to be between 0 and 9. Does that make sense? So we can make a little table: 1^2 2^2 3^2 4^2 1^2 2 5 10 17 2^2 5 8 13 20 3^2 10 13 18 25 4^2 17 20 25 32 I've stopped at 4^2, but if you continue up to 9^2, you'll find all the possibilities. Note that the table is symmetric, because addition is commutative. So you really only have to fill in half of it, i.e., 1^2 2^2 3^2 4^2 1^2 2 5 10 17 2^2 8 13 20 3^2 18 25 4^2 32 Can you take it from here? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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