Select the set in which the numbers are related in the same way as are the numbers of the following set.

(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding/deleting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is NOT allowed)

(7, 3, 43)
(5, 2, 21)

1. (11, 5, 111)
2. (15, 6, 34)
3. (13, 23, 131)
4. (12, 3, 43)

Question

Select the set in which the numbers are related in the same way as are the numbers of the following set.

(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding/deleting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is NOT allowed)

(7, 3, 43)
(5, 2, 21)

1. (11, 5, 111)
2. (15, 6, 34)
3. (13, 23, 131)
4. (12, 3, 43)

Accepted Answer

**Solution:**

1. Analyze the first given set (7, 3, 43):
   - Check if the third number relates to the first two by a formula.
   - Try the formula: $a^2 - b = c$
   - Calculate: $7^2 - 3 = 49 - 3 = 46$ (not 43)

2. Try another formula: $a^2 + b = c$
   - Calculate: $7^2 + 3 = 49 + 3 = 52$ (not 43)

3. Try: $a 	imes b + a = c$
   - Calculate: $7 	imes 3 + 7 = 21 + 7 = 28$ (not 43)

4. Try: $a 	imes b + b = c$
   - Calculate: $7 	imes 3 + 3 = 21 + 3 = 24$ (not 43)

5. Try: $a^2 - b^2 = c$
   - Calculate: $7^2 - 3^2 = 49 - 9 = 40$ (not 43)

6. Try: $a^2 + b^2 = c$
   - Calculate: $49 + 9 = 58$ (not 43)

7. Try: $a^2 - b + 1 = c$
   - Calculate: $49 - 3 + 1 = 47$ (not 43)

8. Try: $a^2 - b - 5 = c$
   - Calculate: $49 - 3 - 3 = 43$ (matches)

Check with second set (5, 2, 21):
- Calculate: $5^2 - 2 - 2 = 25 - 2 - 2 = 21$ (matches)

So the formula is:

$$c = a^2 - b - b$$

or equivalently,

$$c = a^2 - 2b$$

Check options:

- Option 1: (11, 5, 111)
  - Calculate: $11^2 - 2 	imes 5 = 121 - 10 = 111$ (matches)

- Option 2: (15, 6, 34)
  - Calculate: $15^2 - 2 	imes 6 = 225 - 12 = 213$ (not 34)

- Option 3: (13, 23, 131)
  - Calculate: $13^2 - 2 	imes 23 = 169 - 46 = 123$ (not 131)

- Option 4: (12, 3, 43)
  - Calculate: $12^2 - 2 	imes 3 = 144 - 6 = 138$ (not 43)

**Answer:** Option 1 (11, 5, 111) satisfies the same relationship.

Answer

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