Solution:

Clearly, the 40 posters should be arranged in a rectangular array, and there are four ways to do so

For a 1 x 40 array, the number of nails required is  (1 + 1 ) x (40 + 1 ) = 82

For a 2 x 20 array, the number of nails required is ( 2 +1 ) x (20 + 1 ) = 63

For a 4 x 10 array, the number of nails required is  ( 4 + 1 ) x ( 10 + 1 ) = 55

For a 5 x 8 array, the number of nails required is  ( 5 + 1 ) x ( 8 + 1 ) = 54

Hence the minimum number of nails required is 54