What is the largest prime factor of $1337$?
To find the prime factors of $1337$, I can use a method called trial division, where I try to divide $1337$ by the smallest prime numbers until I get a quotient of 1. The smallest prime number is 2, but $1337$ is not divisible by 2, because it is odd. The next smallest prime number is 3, but $1337$ is not divisible by 3, because the sum of its digits is not divisible by 3. The next smallest prime number is 5, but $1337$ is not divisible by 5, because it does not end in 0 or 5. The next smallest prime number is 7, and $1337$ is divisible by 7, because 7 times 191 is 1337. So 7 is a prime factor of $1337$, and the other factor is 191. But is 191 also a prime factor, or can it be further factored? To check, I can continue the trial division with the next prime numbers: 11, 13, 17, 19, etc. None of these numbers divide 191 evenly, so 191 is also a prime factor of $1337$. Therefore, the prime factorization of $1337$ is $7 \times 191$. The largest prime factor of $1337$ is 191. 