Question 1205514
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Let x be the number exactly half-way between the two unknown positive integer numbers.


Since the difference between the numbers is 5, it is clear that the numbers are (x-2.5) and (x+2.5).


Then their product is (x-2.5)*(x+2.5) = x^2 - 2.5^2 = x^2 - 6.25.


It gives us this equation for x

    x^2 - 6.25 = 84.


From this equation

    x^2 = 84 + 6.25 = 90.25.


From this point, we find mentally  x = {{{sqrt(90.25)}}} = 9.5.


Then the lesser of two integer positive numbers is  9.5-2.5 =  7,
while the greater of these two numbers is           9.5+2.5 = 12.


<U>ANSWER</U>.  The numbers are 7 and 12.
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Solved (mentally).