Three of the following number-pairs are alike in some manner and hence form a group. Which number-pair does not belong to that group? (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/subtracting/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.)

Correct — C. Rule: second number = n × (n + 1) — the number multiplied by its own successor.

12 × 13 = 156
21 × 22 = 462
17 × 18 = 306

For 14 the rule wants 14 × 15 = 210, but the pair prints 196, which is 14 × 14 = 14².

A square where the others are a product of consecutive numbers — so option (c) is the pair that does not belong.

Why the others are wrong

• (a)
  12 – 156 — 12 – 156 obeys the rule exactly: 12 × 13 = 156. It sits inside the group, so it cannot be the pair that leaves it.
• (b)
  21 – 462 — 21 – 462 obeys the rule: 21 × 22 = 462. The numbers are the largest on offer, which draws the eye, but size is not the relation being tested.
• (d)
  17 – 306 — 17 – 306 obeys the rule: 17 × 18 = 306. 17 is the only prime among the first numbers, which tempts a guess, yet the multiplication still works.

An odd-one-out on number pairs asks you to find the single relation that binds three pairs, then name the fourth.

Test cheap relations in a fixed order: difference, ratio, square, cube, then n(n + 1) and n² ± n.

Here 156, 462 and 306 are pronic numbers, each the product of two consecutive whole numbers, while 196 is a perfect square. The two families sit close together — 14 × 15 is 210 against 14² = 196 — which is the whole difficulty.

The note under the question bars digit-splitting, so 156 is one hundred and fifty-six, never a 1, a 5 and a 6.

Key facts

• A number of the form n(n + 1) is called pronic, and every pronic number is even.
• 12 × 13 = 156, 17 × 18 = 306 and 21 × 22 = 462.
• 196 = 14², so 14 – 196 is a square relation rather than a pronic one.
• The printed note bars breaking a number into digits, so operations act on the whole number.

Common traps

• Spotting that 196 = 14² and stopping, without checking what the other three share
• Choosing 21 – 462 because those numbers are the biggest
• Splitting the numbers into digits, which the question's own note forbids

No directly related past PYQ was found.