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Let x be the smaller odd number, then the greater one can be expressed in terms of x as x + 2.
As the sum of the squares of two cinsecutive positive odd numbers is 394, we can setup an equation:
x^2 + (x + 2)^2 = 394
Solving the quadratic equation for x, we have:
x^2 + x^2 + 4x + 4 = 394
2x^2 + 4x + 4 = 394
2x^2 + 4x - 390 = 0
x^2 + 2x - 195 = 0
(x + 15)(x - 13) = 0
So x = -15 or x = 13
As x is positive number, we reject the solution x = -15.
So the only solution is x = 13.