<< what is the largest prime number? >>
The largest prime number (if there is one) would not be the solution to this problem anyways, because the problem allows for a linear combination of multiple factors (6, 9, and 20).
I can quickly show that any number greater than 100 is attainable. If you look at the multiples of 9 up to 90, and look at the one's digits, all digits 0-9 are produced. So from this, any number greater than 100 can also be produced by taking each multiple of 9 and then adding multiples 20 or 30 (6*5).
101 = 81 + 20
102 = 72 + 30
103 = 63 + 20 * 2
104 = 54 + 20 + 30
105 = 45 + 20 * 3
And so on... then, looking at all the numbers less than 100, it doesn't take long to find that 43 is the greatest number with no combination.