Is 209 a prime number?

Steps to Solve

1. Check for divisibility by small prime numbers

   First, test if 209 is divisible by the smallest prime numbers: 2, 3, 5, 7, 11, and so on.

   209 is odd, so it is not divisible by 2.

   The sum of the digits of 209 is $2 + 0 + 9 = 11$, which is not divisible by 3, so 209 is not divisible by 3.

   The last digit of 209 is not 0 or 5, so it is not divisible by 5.

   Next, check for divisibility by 7:

   $$ 209 \div 7 = 29.857 $$

   This is not an integer, so 209 is not divisible by 7.

   Next, check for divisibility by 11:

   If the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is a multiple of 11, then the number is divisible by 11.

   $$ 2 - 0 + 9 = 11 $$

   Since 11 is divisible by 11, 209 is divisible by 11.

2. Determine divisibility by other factors

   Since 209 is divisible by 11 (a prime number), and the result is:

   $$ 209 \div 11 = 19 $$

   We need to check if 19 is also a prime number. Since 19 is itself a prime number and 209 equals $11 \times 19$, we conclude that 209 has divisors other than 1 and 209.

3. Conclusion

   Since 209 has divisors other than 1 and itself, it is not a prime number.