18 = 2*3*3

So to avoid getting a product divisible by 18, she could either 
avoid multiples of 2 or 3.  There are fewer multiples of 3, so she
would write down more numbers by avoiding the multiples of 3.

The multiples of 3 not exceeding 100 are 

3,6,9,...,99.

To see how many they are, divide those all by 3

1,2,3,...,33.

So, there are 33 multiples of 3. To exclude them, 100-33 = 67.  

However she can also write ONE multiple of 3 (that isn't a multiple
of 9), without getting a multiple of 18, so that's one more she could 
write, making her list contain 68 numbers.

Answer: 68.  She might as well include 3 as her ONE multiple of 3
(that isn't a multiple of 9), so her list could go: 

1*2*3*4*5*7*8*10*11*13*...95*97*98*100.

Edwin