GATE CS 2014 Set 2 Q49 — Sets, Relations and Functions

To find the number of distinct positive integral factors (divisors) of $2014$, we first determine its prime factorization.

Let's divide $2014$ by the smallest prime numbers:

• $2014$ is even: $$\frac{2014}{2} = 1007$$
• Now, we check for prime factors of $1007$. Testing small primes ($3, 5, 7, 11, 13, 17, 19, \dots$): $$\frac{1007}{19} = 53$$
• Since $53$ is a prime number, the prime factorization of $2014$ is: $$2014 = 2^1 \times 19^1 \times 53^1$$

The total number of distinct positive factors of a number $N = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k}$ is given by the formula:

Applying this formula for $2014$:

The $8$ distinct positive factors are: $1, 2, 19, 38, 53, 106, 1007,$ and $2014$.