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When you say " the sum of the squares of each number is 153", I take to mean "the sum of the squares of both numbers is 153". Based on that assumption, then we have:

x - y =9................................(1)

x^2 + y^2 = 153..................(2)

From (1) above: x =9 + y.   sub this into (2) above:

(9 + y)^2 + y^2 = 153, solve for y

y^2 + 18y + 81 + y^2 =153

2y^2 + 18y + 81 - 153 = 0

2y^2 + 18y - 72 = 0 factor:

2(y - 3)(y + 12) = 0   divide both sides by 2

(y - 3)(y+ 12) = 0

y = 3     or    y = -12 and:

x =12    or    x =- 3

x*y =3*12 =36  or -3*-12 =36

Here is another method:  

Let the numbers be x and y    

\( (x-y)^2=x^2-2xy+y^2\\ \text{But we are told that } x-y=9 \text{ and that } x^2+y^2=153\\ so\\ (9)^2=153-2xy\\ 81-153=-2xy\\ -72=-2xy\\ xy=36\)

So the product of the 2 numbers is 36.