Select the odd group of numbers. Note: Perform operations on each whole number…
2022
Select the odd group of numbers. Note: Perform operations on each whole number without splitting it into its constituent digits. For example, you may add to, subtract from, or multiply 13 as a whole, but you must not split 13 into 1 and 3 for separate operations.
- A.
(153 – 151 – 147)
- B.
(171 – 169 – 167)
- C.
(183 – 181 – 178)
- D.
(201 – 198 – 197)
Attempted by 4 students.
Show answer & explanation
Correct answer: A
Concept
In an odd-one-out problem, apply one consistent operation to every group and compare the same resulting property.
A number is coprime to 6 precisely when it is divisible by neither 2 nor 3. After adding the three whole numbers in each group, compare the greatest common divisor of each sum with 6.
Application
Group | Sum | Division by 6 and common factor |
|---|---|---|
(153 – 151 – 147) | 153 + 151 + 147 = 451 | 451 = 6 × 75 + 1; gcd(451, 6) = 1 |
(171 – 169 – 167) | 171 + 169 + 167 = 507 | 507 = 6 × 84 + 3; gcd(507, 6) = 3 |
(183 – 181 – 178) | 183 + 181 + 178 = 542 | 542 = 6 × 90 + 2; gcd(542, 6) = 2 |
(201 – 198 – 197) | 201 + 198 + 197 = 596 | 596 = 6 × 99 + 2; gcd(596, 6) = 2 |
Cross-check
The last three sums have a common factor greater than 1 with 6: either 3 or 2. The sum 451 alone is divisible by neither 2 nor 3, so its greatest common divisor with 6 is 1.
Result
Only the group (153 – 151 – 147) produces a sum coprime to 6, so it is the odd group.