Prime or Composite: Analyzing the Number 19

Is the number equal to $n$ prime or composite?

$n=19$

To determine if the number 19 is prime, follow these steps:

• Step 1: Check if the number is greater than 1. Since $19 > 1$, proceed to the next step.
• Step 2: Identify potential divisors for 19 by considering integers from 2 up to $\lfloor \sqrt{19} \rfloor$.

The square root of 19 is approximately 4.36, and thus we test divisibility by integers 2, 3, and 4.

• 19 divided by 2: The quotient is not an integer (it gives 9.5).
• 19 divided by 3: The quotient is not an integer (it gives 6.333...).
• 19 divided by 4: The quotient is not an integer (it gives 4.75).

None of these divisions result in an integer, meaning 19 has no divisors other than 1 and 19 itself.

Therefore, the number 19 is prime.