The correct answer is Option (3) → (95, 112)
To determine which pair of numbers is relatively prime (also known as co-prime), we need to check if their Greatest Common Divisor (GCD) is 1. If the GCD is 1, the numbers share no common factors other than 1.
Let's test each pair:
1. (371, 159)
- Factors of 159: $3 \times 53$
- Does 371 divide by 3? No (sum of digits $3+7+1=11$).
- Does 371 divide by 53? Yes ($53 \times 7 = 371$).
- GCD = 53. (Not relatively prime)
2. (407, 259)
- Factors of 407: $11 \times 37$ (Using the 11s rule: $4+7=11$).
- Does 259 divide by 11? No ($2-5+9=6$).
- Does 259 divide by 37? Yes ($37 \times 7 = 259$).
- GCD = 37. (Not relatively prime)
3. (95, 112)
- Factors of 95: $5 \times 19$.
- Factors of 112: $2 \times 2 \times 2 \times 2 \times 7 = 16 \times 7$.
- These two numbers share no common factors.
- GCD = 1. (Relatively prime)
4. (57, 123)
- Check for divisibility by 3 (Sum of digits):
- $5 + 7 = 12$ (Divisible by 3)
- $1 + 2 + 3 = 6$ (Divisible by 3)
- Both are divisible by 3 ($57 = 3 \times 19$ and $123 = 3 \times 41$).
- GCD = 3. (Not relatively prime)
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