How did I get a solution?
First, I found the least common denominator of the numbers between 2 and 9.
To get this, I found the product of 5, 7, 8 and 9=2520. I ignored 2, 3, 4 and 6 because 8 is a multiple of 4 and 2 (which will divide into everything that 8 divides into), and 6 divides into 8×9.
Then I wrote down 2521 and a set of numbers following that.
Next, I crossed off all the numbers which are the sum of 2520 and the numbers 2-9 and their multiples. So between 2521 and about 2540 I was left with:
2521, 2531, 2533, 2537, 2539.
Next, I crossed off the 3 prime numbers in this set leaving:
2533 and 2537.
So I could have also picked 2533=17x149.