Difference of Squares of Two IntegersDate: 7/2/96 at 8:2:7 From: Anonymous Subject: Difference of Squares of Integers 1. Which positive integers can be written as the difference of the squares of two integers? Explain your reasoning. I know that the numbers that are squared cannot be equal, so I came up with this: n^2 - (n-x)^2 = a positive integer. The smallest number that will work is 3 (2^2 - 1^2 = 3). Continuing sequentially, the differences turn out to be odd numbers. If I try non-sequential numbers, however, the difference can be even. (6^2 - 4^2 = 20) I have an equation, but don't know how to make it work. Can you help? Thanks, Nancy Date: 7/2/96 at 10:49:52 From: Doctor Anthony Subject: Re: Difference of Squares of Integers Any expression of the form a^2 - b^2 can be factorized as (a-b)(a+b). It follows that if a number can be factorized in this way, then it can be expressed as a difference of squares. The example you gave could be worked as follows: 20 = 10*2 So a+b = 10 a-b = 2 --------- add 2a = 12 so a = 6 subtract 2b = 8 so b = 4 You will note from this working that the sum of the factors must be even, since they will equal 2a (and the difference will equal 2b). So if a number can be factorized into two even or two odd factors, then it can be expressed as a difference of squares. Example. 35 = 7*5 a+b = 7 a-b = 5 --------- 2a = 12 and a = 6 b = 1 and it is true 35 = 36-1 = 6^2 - 1^2 Example 63 = 9*7 a+b = 9 a-b = 7 2a = 16 a=8 b=1 and 8^2-1^2 = 63 Example 60 = 10*6 a+b = 10 a-b = 6 2a = 16 a=8 b=2 and 8^2-2^2 = 60 So the general rule is that provided a number can be factorized with two even or two odd factors, then it can be represented as a difference of squares. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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