Solutions: Section 5.2

Page 113

Q. Primes and Composites from 21 to 30.

Numbers: 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.

• Primes: 23, 29 (Count: 2).
• Composites: 21, 22, 24, 25, 26, 27, 28, 30 (Count: 8).

Page 114: Figure it Out

Q1. Is there any other even prime besides 2?

Answer: No. Reason: Any other even number is divisible by 2 (has factors 1, 2, itself…), making it composite.

Q2. Smallest and largest difference between successive primes (1-100).

• Smallest: 3 - 2 = 1. (This is the only pair with diff 1).
• Largest: Look at gaps.
  • 89 to 97. Difference = 8.

Q4. Which are prime: 23, 51, 37, 26?

• 23: Prime.
• 51: Composite ($3 \times 17$).
• 37: Prime.
• 26: Composite ($2 \times 13$). Answer: 23 and 37.

Q6. Pairs like 13 and 31 (Prime with same digits).

Answer:

• 17 and 71
• 37 and 73
• 79 and 97

Q7. Seven consecutive composite numbers (1-100).

We need a large gap between primes. Look at the 90s: 90, 91 ($7 \times 13$), 92, 93 ($3 \times 31$), 94, 95, 96. Next prime is 97. Answer: 90, 91, 92, 93, 94, 95, 96.

Q8. Twin Primes (Diff 2).

Answer: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).