Question

The number of positive divisors of $2^5{3}^6{7}^3$ is

• A. $14$
• B. $167$
• C. $168$
• D. $210$

Answer 1

Answer: (C)

$168$

All the positive divisors will have a prime factorization comprised of some subset of the prime factors of the number given.
You can have anything between zero to five $2^{\prime }$s in the divisor's prime factorization, that is $6$ ways to take the prime factor $2$, one extra for not taking $2$ at all $(2$ raised to $0).$

Using the same logic for the other prime factors, and multiplying the number of possibilities for each:
$(5+1)(6+1)(3+1)=168$