GATE CS 2015 Set 2 Q9 — Sets, Relations and Functions

To find the number of divisors of $2100$, we first find its prime factorization.

Divide $2100$ into its prime factors:

Any positive divisor of $2100$ is of the form $2^a \times 3^b \times 5^c \times 7^d$, where:

• $a \in \{0, 1, 2\}$ (3 choices)
• $b \in \{0, 1\}$ (2 choices)
• $c \in \{0, 1, 2\}$ (3 choices)
• $d \in \{0, 1\}$ (2 choices)

Using the formula for the number of divisors of a number $n = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}$:

Substituting the exponents of the prime factors of $2100$:

Thus, the total number of divisors of $2100$ is 36.