Solutions: Section 5.3

Page 115: Safe Pairs

Q. Check if these pairs are safe (Co-prime).

“Safe” means Jumpy cannot land on both with a jump size > 1. This means they have no common factors > 1.

a. 15 and 39

• Factors of 15: 1, 3, 5.
• Factors of 39: 1, 3, 13.
• Common: 3. Not Safe.

b. 4 and 15

• Factors of 4: 1, 2, 4.
• Factors of 15: 1, 3, 5.
• Common: 1. Safe (Co-prime).

c. 18 and 29

• 29 is prime. 18 is not a multiple of 29.
• Safe (Co-prime).

d. 20 and 55

• Both divisible by 5. Not Safe.

Page 116

Q. Which pairs are co-prime?

a. 18 and 35:

• $18 = 2 \times 3^2$
• $35 = 5 \times 7$
• No common factors. Co-prime.

b. 15 and 37:

• 37 is prime. Co-prime.

c. 30 and 415:

• Both end in 0 or 5. Divisible by 5. Not Co-prime.

d. 17 and 69:

• 17 is prime. $17 \times 4 = 68$. 69 is not a multiple. Co-prime.

e. 81 and 18:

• Both divisible by 9. Not Co-prime.