The number needs to be less than 13 × 52 = 676. The highest power of 13 in 676! is 56. The power of 13 in the smallest such number needs to be exactly 52. If we subtract 13 × 3 = 39 from 676, we get 637. The number 637! will be the smallest number of type N! that is completely divisible by 1352. The sum of the digits of 637 is 16.