let x and y represent the two whole numbers

The product of two whole numbers is 360

xy = 360

y = 360/x

their sum is less than 100:

x + y < 100

plug y = 360/x into the last equation and simplify:

x^2 - 100x + 360 < 0

if we draw the graph for y = x^2 - 100x + 360 we see that for every integer x in the interval [4, 96] the inequality holds

click here to see the graph of the equation:

Click to see all the steps

thus the possible solutions are all the number in the interval [4, 96] that divide 360 (since y = 360/x is integer:

x = 4, y = 360/4 = 90

x = 5, y = 360/5 = 72

x = 6, y = 360/6 = 60

and so on.