Is 349 prime?

Steps to Solve

1. Identify the definition of a prime number

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, a prime number has exactly two distinct positive divisors: 1 and itself.

2. Check for factors of 349

To determine if 349 is prime, we need to check if it can be evenly divided by any integers other than 1 and 349. We do this by testing divisibility by prime numbers up to the square root of 349. The square root of 349 is approximately 18.68, so we’ll check for divisibility by the prime numbers less than or equal to 18: 2, 3, 5, 7, 11, 13, 17.

3. Testing divisibility by each prime number

• Divisibility by 2: 349 is an odd number, so it is not divisible by 2.

• Divisibility by 3: The sum of the digits of 349 is $3 + 4 + 9 = 16$, which is not divisible by 3.

• Divisibility by 5: 349 does not end in 0 or 5, so it is not divisible by 5.

• Divisibility by 7: Dividing $349 \div 7 \approx 49.857$, which is not an integer.

• Divisibility by 11: Dividing $349 \div 11 \approx 31.727$, which is not an integer.

• Divisibility by 13: Dividing $349 \div 13 \approx 26.077$, which is not an integer.

• Divisibility by 17: Dividing $349 \div 17 \approx 20.529$, which is not an integer.

4. Conclusion about the primality of 349

Since 349 is not divisible by any prime numbers up to its square root, we conclude that 349 has no divisors other than 1 and itself. Therefore, it is a prime number.