Chapter 9.5, Problem 24ES

The number 42 has the prime factorization $2\cdot 3\cdot 7$ . Thus 42 can be written in four ways as a product of two positive integer factors (without regard to the order of the factors): $1\cdot 42,2\cdot 21,3\cdot 14,$ and $6\cdot 7$ . Answer a—d below without regard to the order of the factors.
a. List the distinct ways the number 210 can be written as a product of two positive integer factors.
b. If $n=p_1{p}_2{p}_3{p}_4$ , where the $p_i$ are distinct prime numbers, how many ways can n be written as a product of two positive integer factors?
c. If $n=p_1{p}_2{p}_3{p}_4{p}_5$ , where the $p_i$ are distinct prime numbers, how many ways can n he written as a product of two positive integer factors?
d. If $n=p_1{p}_2\dots p_k$ , where the $p_i$ are distinct prime numbers, how many ways can n be written as a product of two positive integer factors?