Use the test for primality to determine which of the folowing natural numbers are prime or composite. \( \begin{array}{lll}\text { (a) } 173 & \text { (b) } 338 & \text { (c) } 748 \\ \text { (d) } 1774\end{array} \) \( \begin{array}{l}\text { (a) Is } 173 \text { prime? If not, state the smallest prime number by which } 173 \text { is divisible. Select the } \\ \text { correct choice below, and if necessary, fill in the answer box within your choice. } \\ \text { A. The number is not prime. The smallest prime number that it is divisible by is } \\ \text { B. The number is prime. }\end{array} \)

To determine whether 173, 338, 748, and 1774 are prime or composite, we can test each number for divisibility by prime numbers up to their square roots. For 173, the square root is approximately 13.15, so we check divisibility by \(2, 3, 5, 7, 11,\) and \(13\): - \(2\): Not divisible since 173 is odd. - \(3\): Sum of digits (1 + 7 + 3 = 11) is not divisible by 3. - \(5\): Last digit is not 0 or 5. - \(7\): \(173 \div 7 \approx 24.71\), not an integer. - \(11\): \(173 \div 11 \approx 15.73\), not an integer. - \(13\): \(173 \div 13 \approx 13.31\), not an integer. Since 173 is not divisible by any of these primes, it is a prime number. Therefore, the answer is: B. The number is prime. Now, let's move on to 338. The square root is about 18.38, so we check up to \(17\): - \(2\): 338 is even, so it is divisible by 2 (\(338 \div 2 = 169\)). Since 338 is divisible by 2, it is not a prime number. The smallest prime number it is divisible by is: A. The number is not prime. The smallest prime number that it is divisible by is 2. Now for 748, with a square root of about 27.36, we check \(2, 3, 5, 7, 11, 13, 17, 19, 23\): - \(2\): 748 is even, so it is divisible by 2 (\(748 \div 2 = 374\)). So, 748 is also not prime. The smallest prime number that it is divisible by is: A. The number is not prime. The smallest prime number that it is divisible by is 2. Finally, for 1774, the square root is about 42.05, so we check primes up to \(41\): - \(2\): 1774 is even, so it is divisible by 2 (\(1774 \div 2 = 887\)). Thus, 1774 is not prime either. The smallest prime number it is divisible by is: A. The number is not prime. The smallest prime number that it is divisible by is 2. In summary: - (a) 173: B. The number is prime. - (b) 338: A. The number is not prime. The smallest prime number that it is divisible by is 2. - (c) 748: A. The number is not prime. The smallest prime number that it is divisible by is 2. - (d) 1774: A. The number is not prime. The smallest prime number that it is divisible by is 2.