The two positive integers:

Let x = one of them

And y = the other, and greater than x.

So,

y -x = 12

y = x +12 ---------**

Their Harmonic Mean,

H = 2 / [1/x +1/(x+12)] = 5

2 / [1/x +1/(x+12)] = 5

2/5 = 1/x +1/(x+12)

Clear the fractions, multiply both sides by 5(x)(x+12),

2(x)(x+12) = 1(5)(x+12) +1(5)(x)

2x^2+24x = 5x +60 +5x

2x^2 +24x -10x -60 = 0

2x*2 +14x -60 = 0

x^2 +7x -30 = 0

(x +10)(x -3) = 0

x = -10 or 3

Positive integer, so x = 3

y = 3 +12 = 15

Check,

2 / [1/3 +1/15] =? 5

2 / [(15 +3)/(3*15)] =? 5

2 / [18 / 45] =? 5

2(45/18) =? 5

90/18 =/ 5

5 =? 5

Yes, so, OK.

Therefore, the two positive integers are 3 and 15. --------answer.