What is the smallest number divisible by 2345678 and 9?

To get at this figure we can come at the problem from the other side.

If we work dowwards from 9 and add the prime factors of each number to a pool, in there are not in it already, then at the end all the numebrs in the pool can be multiplied together.

9 - factors 3, 3 ....................................... pool (3, 3)

8 - factors 2, 2, 2, .................................. pool (3, 3, 2, 2, 2)

7 - no factors ......................................... pool (3, 3, 2, 2, 2, 7)

6 - factors 3, 2 (both already in pool) ........ pool (3, 3, 2, 2, 2, 7)

5 - no factors ......................................... pool (3, 3, 2, 2, 2, 7, 5)

4 - factors 2, 2 (both already in pool)......... pool (3, 3, 2, 2, 2, 7, 5)

3 - no factors - already in pool .................. pool (3, 3, 2, 2, 2, 7, 5)

2 - no factors - already in pool .................. pool (3, 3, 2, 2, 2, 7, 5)

So if we now multiply 3 x 3 x 2 x 2 x 2 x 7 x 5, we get 2.520

2,520 divided by 2 = 1,260

2,520 divided by 3 = 840

2,520 divided by 4 = 630

2,520 divided by 5 = 504

2,520 divided by 6 = 420

2,520 divided by 7 = 360

2,520 divided by 8 = 315

2,520 divided by 9 = 280