Puzzles

What is the largest number of factors which a number less than a million has?

Can you place the digits 1 to 9 in the boxes so that the three digit numbers formed in the top, middle and bottom rows are multiples of 17, 25 and 9 (respectively); and the three digit numbers in the left, middle and right columns are multiples of 11, 16 and 12 (respectively)?

Consider three-digit integers \(N\) such that:

(a) \(N\) is not exactly divisible by 2, 3 or 5.

(b) No digit of \(N\) is exactly divisible by 2, 3 or 5.

How many such integers \(N\) are there?