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

$\sqrt{21} \approx 4.58$.

• 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 ($21 = 3 \times 7$). Since 21 has factors other than 1 and itself, it is a composite number. The answer is $\boxed{\text{Composite}}$.

• b) 29

$\sqrt{29} \approx 5.38$.

• 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 $\boxed{\text{Prime}}$.

• c) 71

$\sqrt{71} \approx 8.42$.

• 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.
• $71 \div 7 = 10$ 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 $\boxed{\text{Prime}}$.

• d) 97

$\sqrt{97} \approx 9.85$.

• 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.
• $97 \div 7 = 13$ 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 $\boxed{\text{Prime}}$.

• e) 111

$\sqrt{111} \approx 10.53$.

• 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 ($111 = 3 \times 37$). Since 111 has factors other than 1 and itself, it is a composite number. The answer is $\boxed{\text{Composite}}$.

• f) 143

$\sqrt{143} \approx 11.96$.

• 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.
• $143 \div 7 = 20$ with a remainder of 3. So, 143 is not divisible by 7.
• $143 \div 11 = 13$. So, 143 is divisible by 11 ($143 = 11 \times 13$). Since 143 has factors other than 1 and itself, it is a composite number. The answer is $\boxed{\text{Composite}}$.