Select the option in which the numbers share the same relationship as that shared by the given pair of numbers: 17 : 67 and 21 : 79. Operations should be performed on the whole numbers without breaking them into digits.

Question

Accepted Answer

Step 1: Analyze the given pairs:
- Pair 1: 17 : 67
- Pair 2: 21 : 79

Step 2: Check the difference between the numbers in each pair:
- 67 - 17 = 50
- 79 - 21 = 58

Step 3: The difference is not the same, so check for another pattern.

Step 4: Check if the second number is a prime number:
- 67 is prime
- 79 is prime

Step 5: Check if the first number is prime:
- 17 is prime
- 21 is not prime

Since 21 is not prime, the pattern is not about both numbers being prime.

Step 6: Check if the second number is the next prime number after the first number:
- Next prime after 17 is 19, but second number is 67, so no.

Step 7: Check if the second number is the sum of the first number and a prime number:
- 17 + 50 = 67 (50 is not prime)
- 21 + 58 = 79 (58 is not prime)

Step 8: Check if the second number is the sum of the first number and the next prime number after the first number:
- Next prime after 17 is 19, 17 + 19 = 36 (not 67)
- Next prime after 21 is 23, 21 + 23 = 44 (not 79)

Step 9: Check if the second number is the sum of the first number and the square of the first number's digit sum:
- Sum of digits of 17 = 1 + 7 = 8; 8^2 = 64; 17 + 64 = 81 (not 67)
- Sum of digits of 21 = 2 + 1 = 3; 3^2 = 9; 21 + 9 = 30 (not 79)

Step 10: Check if the second number is the sum of the first number and the first number multiplied by 3:
- 17 + (17*3) = 17 + 51 = 68 (close to 67)
- 21 + (21*3) = 21 + 63 = 84 (not 79)

Step 11: Check if the second number is the first number multiplied by 4 minus 1:
- 17*4 - 1 = 68 - 1 = 67
- 21*4 - 1 = 84 - 1 = 83 (not 79)

Step 12: Check if the second number is the first number multiplied by 4 minus 3:
- 17*4 - 3 = 68 - 3 = 65 (not 67)
- 21*4 - 3 = 84 - 3 = 81 (not 79)

Step 13: Check if the second number is the first number multiplied by 4 minus 1 for the first pair and multiplied by 3 plus 16 for the second pair:
- 17*4 - 1 = 67
- 21*3 + 16 = 63 + 16 = 79

This suggests the pattern is not consistent.

Step 14: Check if the second number is the first number multiplied by 3 plus 16:
- 17*3 + 16 = 51 + 16 = 67
- 21*3 + 16 = 63 + 16 = 79

This fits both pairs.

Step 15: Apply this to options:
- Option 1: 33*3 + 16 = 99 + 16 = 115 (not 110)
- Option 2: 34*3 + 16 = 102 + 16 = 118 (matches option 2's second number)
- Option 3: 42*3 + 16 = 126 + 16 = 142 (not 100)
- Option 4: 23*3 + 16 = 69 + 16 = 85 (not 69)

Step 16: Check option 2's second number is 118, which matches calculation.

Step 17: But option 2's second number is 118, which matches the pattern.

Step 18: Re-examine option 1:
- 33*3 + 16 = 115, but option 1's second number is 110, so no.

Step 19: So option 2 is correct.

Final answer: Option 2 (34 : 118) shares the same relationship as the given pairs.

Answer

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