Is 397 a prime number?

Steps to Solve

1. Identify potential factors of 397

To determine if 397 is prime, we start by identifying potential factors. We only need to check for divisibility by prime numbers up to the square root of 397. Calculate the square root:

$$ \sqrt{397} \approx 19.93 $$

This means we need to check for factors up to 19.

2. Check divisibility by small prime numbers

We will check if 397 is divisible by the prime numbers less than or equal to 19: 2, 3, 5, 7, 11, 13, 17, and 19.

• Divisibility by 2: 397 is odd, so it is not divisible by 2.

• Divisibility by 3: Sum of digits of 397 is $3 + 9 + 7 = 19$, which is not divisible by 3.

• Divisibility by 5: 397 does not end in 0 or 5, so it is not divisible by 5.

• Divisibility by 7: Calculate $397 \div 7 \approx 56.71$, not an integer.

• Divisibility by 11: Calculate $397 \div 11 \approx 36.09$, not an integer.

• Divisibility by 13: Calculate $397 \div 13 \approx 30.54$, not an integer.

• Divisibility by 17: Calculate $397 \div 17 \approx 23.35$, not an integer.

• Divisibility by 19: Calculate $397 \div 19 \approx 20.89$, not an integer.

3. Conclude the primality of 397

Since 397 is not divisible by any of the prime numbers we tested, we conclude that 397 has no factors other than 1 and itself.