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)**