Determine whether each of these integers is prime: a) 21 b) 29 c) 71

Here are the step-by-step solutions for the first question. 1. Determine whether each of these integers is prime. To determine if a number is prime, we check for divisibility by prime numbers up to its square root. If no such prime divides the number, it is prime. Otherwise, it is composite. a) 21 Step 1: Find the square root of 21. sqrt(21) ≈ 4.58. Step 2: Check for divisibility by prime numbers less than or equal to 4.58 (i.e., 2, 3). • 21 is not divisible by 2 (it is an odd number). • The sum of the digits of 21 is 2+1=3, which is divisible by 3. So, 21 is divisible by 3 (21 = 3 × 7). Since 21 has factors other than 1 and itself, it is a composite number. The answer is Composite. b) 29 Step 1: Find the square root of 29. sqrt(29) ≈ 5.38. Step 2: Check for divisibility by prime numbers less than or equal to 5.38 (i.e., 2, 3, 5). • 29 is not divisible by 2 (it is an odd number). • The sum of the digits of 29 is 2+9=11, which is not divisible by 3. So, 29 is not divisible by 3. • 29 does not end in 0 or 5. So, 29 is not divisible by 5. Since 29 is not divisible by any prime number up to its square root, it is a prime number. The answer is Prime. c) 71 Step 1: Find the square root of 71. sqrt(71) ≈ 8.42. Step 2: Check for divisibility by prime numbers less than or equal to 8.42 (i.e., 2, 3, 5, 7). • 71 is not divisible by 2 (it is an odd number). • The sum of the digits of 71 is 7+1=8, which is not divisible by 3. So, 71 is not divisible by 3. • 71 does not end in 0 or 5. So, 71 is not divisible by 5. • 71 ÷ 7 = 10 with a remainder of 1. So, 71 is not divisible by 7. Since 71 is not divisible by any prime number up to its square root, it is a prime number. The answer is Prime. d) 97 Step 1: Find the square root of 97. sqrt(97) ≈ 9.85. Step 2: Check for divisibility by prime numbers less than or equal to 9.85 (i.e., 2, 3, 5, 7). • 97 is not divisible by 2 (it is an odd number). • The sum of the digits of 97 is 9+7=16, which is not divisible by 3. So, 97 is not divisible by 3. • 97 does not end in 0 or 5. So, 97 is not divisible by 5. • 97 ÷ 7 = 13 with a remainder of 6. So, 97 is not divisible by 7. Since 97 is not divisible by any prime number up to its square root, it is a prime number. The answer is Prime. e) 111 Step 1: Find the square root of 111. sqrt(111) ≈ 10.53. Step 2: Check for divisibility by prime numbers less than or equal to 10.53 (i.e., 2, 3, 5, 7). • 111 is not divisible by 2 (it is an odd number). • The sum of the digits of 111 is 1+1+1=3, which is divisible by 3. So, 111 is divisible by 3 (111 = 3 × 37). Since 111 has factors other than 1 and itself, it is a composite number. The answer is Composite. f) 143 Step 1: Find the square root of 143. sqrt(143) ≈ 11.96. Step 2: Check for divisibility by prime numbers less than or equal to 11.96 (i.e., 2, 3, 5, 7, 11). • 143 is not divisible by 2 (it is an odd number). • The sum of the digits of 143 is 1+4+3=8, which is not divisible by 3. So, 143 is not divisible by 3. • 143 does not end in 0 or 5. So, 143 is not divisible by 5. • 143 ÷ 7 = 20 with a remainder of 3. So, 143 is not divisible by 7. • 143 ÷ 11 = 13. So, 143 is divisible by 11 (143 = 11 × 13). Since 143 has factors other than 1 and itself, it is a composite number. The answer is Composite.