Chapter 8.6, Problem 20P

a. Suppose that a fraction $\frac{A}{B}$ where A and B are counting numbers has a terminating decimal representation and $\frac{A}{B}$ that is in simplest form. Explain why B must divide a power of 10.

b. Suppose that a fraction $\frac{A}{B}$ where A and B are counting numbers has a terminating decimal representation and $\frac{A}{B}$ that is in simplest form. Show that when B is factored into a product of prime numbers, the only prime numbers that can appear are 2 and 5.

c. Given a counting number B, suppose that when B is factored into a product of prime numbers, the only prime numbers that appear are 2, or 5, or both 2 and 5. Let A be another counting number. Explain why $\frac{A}{B}$ must have a terminating decimal representation.

d. Does $\frac{1}{2^{100}}$ have a repeating or a terminating decimal representation? Explain your answer.

e. Does $\frac{1}{3^{100}}$ have a repeating or a terminating decimal representation? Explain your answer.