Question

Determine whether each of the following numbers is prime or composite. In each case, give a reason for your answer. a. 35 b. 31 c. 51

Short answer

Question: Determine if the following numbers are prime or composite: 35, 31, and 51. Answer: 35 is a composite number, 31 is a prime number, and 51 is a composite number.

Step-by-step solution

Analyze number 35

To determine if 35 is prime or composite, we can seek factors other than 1 and 35. Since 35 is an odd number, it can't be divided by 2. However, knowing our multiplication table, we know that 5 × 7 = 35. This means that 35 has factors other than 1 and itself, namely 5 and 7.

Analyze number 31

To determine if 31 is prime or composite, let's look for factors other than 1 and 31. As 31 is an odd number, it won't be divisible by 2. Let's check through numbers 3, 5, 7, 11, 13, etc., to seek a factor. If no factor is found before reaching a number greater than the square root of 31 (which is around 5.56), we can conclude that 31 is prime. In this case, none of these numbers (3, 5, 7) divide 31 evenly. So, 31 has no factors other than 1 and itself.

Analyze number 51

Lastly, let's determine if 51 is prime or composite. Like before, we look for factors other than 1 and 51. Since 51 is odd, it won't be divisible by 2. Let's now check through numbers like 3, 5, 7, 11, 13, etc. In this case, we find that 3 × 17 = 51, which means 51 has factors other than 1 and itself, namely 3 and 17.

Summarize the Results

After analyzing each number, we can conclude the following: - 35 is a composite number because it has factors other than 1 and itself (5 and 7). - 31 is a prime number because it has no factors other than 1 and itself. - 51 is a composite number because it has factors other than 1 and itself (3 and 17).