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October 19th, 2012, 04:25 AM   #1
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x*y = x+y Question about mathematical proof

Short math-quiz for those interested: find two values x and y that satisfies the equation x*y = x+y where x is an integer and y has one decimal. If you don't want to know the/an answer then DON'T CONTINUE READING.



Question: I don't have a higher math education than an engineering degree, and so I don't really know anything about mathematical proof. It suddenly struck me that 3 and 1,5 satisfies the equation in my subject, but what I am wondering is: is there a way to prove/disprove that there are or aren't any other combinations of integers and one-decimal numbers that also satisfy the equation? I know for integers 2 and 0 satisfy the equation, but if you include an infinite amount of decimals for both x and y, is there an infinite number of solutions to this equation? If so, how does one prove that?
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October 19th, 2012, 06:18 AM   #2
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Re: x*y = x+y Question about mathematical proof

So it's convenient to write variables which are known to be integers, since then any combination of variables and integers via addition, subtraction, or multiplication gives integers. So I will write this as

x * (y/10) = x + (y/10)

which can be rewritten as

xy = 10x + y

which is (for me) much more pretty. Now let's isolate a variable, say y:

xy - y = 10x
y(x - 1) = 10x
y = 10x/(x-1)

x and x-1 have no common factors, so this can be an integer only if x-1 divides 10. The divisors of 10 are 1, 2, 5, and 10, so x can be -9, -4, -1, 0, 2, 3, 6, or 11. You can find the associated values of y by putting x back into the equation.

Once you have those, you can divide y by 10 to get answers to the original problem. Depending on how you interpret the requirement that y has one decimal, you may need to ignore two of these solutions, (2, 2.0) and (0, 0.0).
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October 19th, 2012, 06:20 AM   #3
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Re: x*y = x+y Question about mathematical proof

Nice. Thanks.
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October 19th, 2012, 06:26 AM   #4
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Re: x*y = x+y Question about mathematical proof

Quote:
Originally Posted by ersvale
if you include an infinite amount of decimals for both x and y, is there an infinite number of solutions to this equation? If so, how does one prove that?
By following my above post you can see that the number of solutions depends on the number of (positive or negative) divisors of 10. If you allow a two-decimal number you look at the (positive or negative) divisors of 100 instead. If you allow any finite number of decimals you're looking at A003592 which is infinite (and so there are infinitely many solutions). If you allow infinitely many decimal places -- that is, real numbers -- you get uncountably many solutions, a larger sort of infinity.
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October 19th, 2012, 12:53 PM   #5
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Re: x*y = x+y Question about mathematical proof

Beautiful! That's exactly what I was looking for. Thank you!
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