Difference of Two Cubes
Date: 05/24/2001 at 19:37:03 From: Jackie Subject: A problem in class we have to do The difference of two cubes is 56,765. What two positive integers satisfy this condition? I have tried the formula (_ ^3-56765)^(1/3). In the blank I put 64 and got the integer 59, so I know that is one number, but I don't know what to do next. I did most of it by the guess-and-check method. Please help.
Date: 05/30/2001 at 13:41:44 From: Doctor Toughy Subject: Re: A problem in class we have to do Hi Jackie, Thanks for writing to Dr. Math. It looks like you solved the problem. Good job. One cube equals 64^3 and the other cube equals 59^3. In the future, whenever you're given a problem such as this one where there is no clear formula for solving it and the numbers used are large, it is usually a good first step to create a simpler problem similar to the one given and then determine how to solve the simpler problem. Often if you make a table it's possible to spot patterns using small numbers in your table that you can later use to solve your more difficult problem. Suppose we had the following problem: The difference of two cubes is 279. What two positive integers satisfy this condition? Let a be the larger integer and b the smaller. The equation for this would be: a^3 - b^3 = 279 To solve the problem, we create a table using possible values for a and b. In the first column we have the value a. In the second column we have a^3 - b^3, where b equals a-1. In the third column we have a^3 - b^3, where b equals a-2. In the third column a^3 - b^3 always equals a | a^3 - b^3 | a^3 - b^3 | a^3 - b^3 | (where b = a-1) | (where b = a-2) | (where b = a-3) ____|________________________________________________________ 1 | 1 | | | | | 2 | 7 | 8 | | | | 3 | 19 | 26 | 27 | | | 4 | 37 | 56 | 63 | | | 5 | 61 | 98 | 117 | | | 6 | 91 | 152 | 189 One of the patterns here is that when b = a-1 then a^3 - b^3 = 1 mod 6, when b = a-2 then a^3 - b^3 = 2 mod 6, when b = a-3 then a^3 - b^3 = 3 mod 6 Now the modular value of 279 is 3 mod 6, so we know that a-b must equal 3 mod 6, which can equal 3, 9, 15 ... Let's continue the table where b = a-3. As the values of a and b get larger, do you see a pattern in the increase in the amount of difference between a^3 and b^3, where the difference between a and b is 3? The equation for this difference is difference = 9 * ((a-1)^2 -(a-2)) Now you can follow the same process and solve for a difference of 56,765. Hope this helps. If you have any more questions about this or any other math topics, please write back. - Doctor Toughy, The Math Forum http://mathforum.org/dr.math/
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