In Kakuro (also known as Cross Sums), no digits can be repeated in each sum.  For example, 16 for two boxes must be 9 and 7 (or 7 and 9) and cannot be 8 and 8.

Q:  How many factors of 155^9 are perfect squares and/or perfect cubes?

A: 37 factors of 155^9 are perfect squares and/or perfect cubes.

155 is a composite number composed of 5 x 31.
Therefore
155^9 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 31 x 31 x 31 x 31 x 31 x 31 x 31 x 31 x 31

25 Perfect Squares:
1: 1 is both a perfect square and a perfect cube.
4:  Perfect squares with just 5s = 5^2, 5^4, 5^6, 5^8
4:  Perfect squares with just 31s = 31^2, 31^4, 31^6, 31^8

Combining the 5s and the 31s Perfect Squares
4: 5^2 x 31^ 2, 5^4 x 31^2, 5^6 x 31^2, 5^8 x 31^2 
4: 5^2 x 31^ 4, 5^4 x 31^4, 5^6 x 31^4, 5^8 x 31^4 
4: 5^2 x 31^ 6, 5^4 x 31^6, 5^6 x 31^6, 5^8 x 31^6 
4: 5^2 x 31^8, 5^4 x 31^8, 5^6 x 31^8, 5^8 x 31^8 

12 Perfect Cubes:
Perfect cubes with just 5s = 5^3, 5^6 (already counted above), 5^9  (2)
Perfect cubes with just 31s = 31^3, 31^6 (already counted above), 31^9 (2)

Combining the 5s Perfect Cubes and the 31s Perfect Cubes
5^3 x 31^3, 5^6 x 31^3, 5^9 x 31^3  (3)
5^3 x 31^6, 5^9 x 31^6 (2)
5^3 x 31^9, 5^6 x 31^9, 5^9 x 31^9  (3)

Q: If 6 people take 3 days to dig 8 ditches, working at the same rate how long will it take 4 people to dig 10 ditches?

A: 5.625 days

18 people days to dig 8 ditches = 2.25 people days per ditch

10 ditches = 22.50 people days

22.50 / 4 = 5.625 days.