If anyone needs help with this question then I have an answer for you!
we take the harmonic mean formula 2ac/(a+c), we know the harmonic mean is 20, so we write 2ac/(a+c)=20.
Next we take our bottom denominator and take it to the left side of the equation giving us 2ac=20(a+c)
we simplify the left side of the equation giving us 2ac=20a+20c, and yes there is somthing common here, 2 so lets factor 2 out of our equation,
ac=10a+10c, now let's take 10a+10c to the left side of the equation ac-10a-10c=0, we need a number that we can add onto both sides, we will need to do -10^2=100, let's add 100 to both sides of our equation, ac-10a-10c=100, now we need to find factors of 100, the number of factors 100 has will be the number of pairs, the factors of 100 are 1,2,4,5,10,20,25,50,100, hence giving our answer as 9 pairs.