-3
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It has been shown that every positive integer is the product of four rational numbers whose sum is 0.

Thus:

3 = (242/75) x (-25/6) x (25/22) x (-54/275) while 
0 = (242/75) + (-25/6) + (25/22) + (-54/275).

Find four rational numbers whose product is 10 and whose sum is 0.

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  • $\begingroup$ The source of this problem is doi.org/10.4169/amer.math.monthly.123.10.989. $\endgroup$ – Bernardo Recamán Santos Dec 26 '20 at 17:22
  • $\begingroup$ Is there a reason you're insisting on code formatting for your math? The point of code formatting is to signal to the computer that you're using code, which isn't the case here. $\endgroup$ – bobble Dec 26 '20 at 19:01
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We are looking for rational numbers, so we can search for integers a,b,c,d,x where
a+b+c+d = 0
abcd = 10x^4
The most simple solution (if it exists) would use one 5 (the larger prime factor) and powers of 2.
Since 5 is odd , we must use 1
We can use 5 and 1 with same sign, leading to
-2.-4.5.1 = 10x^4
We can use 5 and 1 with different sign, leading to
-1.-8.5.4 = 10x^4
The second one gives x=2, giving a solution:

-1/2,-8/2,5/2,4/2

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