Classification (Odd One Out)

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Number odd one out

Three or four numbers share a number-theory property; one breaks it. The property bank, in testing order:

Prime vs composite (most frequent)
Perfect square / cube / power
Multiples (of 7, 9, 11, 13...)
Digit properties — digit sum 9, digit product, repeating digits
Parity (odd/even) — usually the second level when all share property 1-3
Divisibility by 11 (alternate digit sums equal)

Two-level questions are rising: all four may be squares, but three are squares of odd numbers. Always check the survivors for a shared sub-property before answering.

📖 Notes 🧩 6 question types ⚡ Formulas 🎯 20 practice Q

Detailed notes

What is number classification?

Four numbers are given. Three share a number property; one does not. You do not need long calculation - you need a list of properties to test in a fixed order.

Everyday example: 2, 4, 6, 9. Three are even; 9 is odd. So 9 is the odd one.

The property list (test in this order)

1. Prime or composite. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...). A composite number has more factors. To test a number up to 200, divide by 2, 3, 5, 7, 11 and 13 only.

Tricky composites that look prime: 51 = 3 × 17, 57 = 3 × 19, 87 = 3 × 29, 91 = 7 × 13, 119 = 7 × 17, 133 = 7 × 19, 143 = 11 × 13, 161 = 7 × 23.

2. Perfect square or cube. Learn squares up to 30² (900) and cubes up to 12³ (1728). A square never ends in 2, 3, 7 or 8.

3. Multiples (divisibility).

By 3 or 9: digit sum divisible by 3 or 9. (45 → 4 + 5 = 9)
By 11: (sum of digits at odd places) − (sum at even places) is 0 or a multiple of 11. For 2849: (2 + 4) − (8 + 9) = −11, so 2849 is divisible by 11.
By 7 and 13: just divide; the numbers are small.

4. Digit property. All four may have the same digit sum or digit product, except one. 138, 234 and 164 all have digit product 24; 326 has 36.

5. Parity (odd/even). Usually a second-level rule: all four are squares, but three are squares of odd numbers.

6. Number forms. Three numbers fit a formula such as $n^2+1$ (50, 65, 82), $n^3-1$ (124, 215, 342) or $n(n+1)$ (30, 42, 56). One does not.

Worked example

Odd one: 79, 83, 89, 91.

All odd, all two-digit - no help.
Prime check: 79, 83, 89 have no divisor from 2, 3, 5, 7. But 91 = 7 × 13.
Answer: 91.

Two-level questions

Sometimes all four numbers pass the first test. Example: 27, 125, 343, 512 are all cubes ( $3^3, 5^3, 7^3, 8^3$ ). Look at the roots: 3, 5, 7 are odd; 8 is even. Answer 512. When the first rule fits all four, always look at the roots or at parity next.

Checking your answer

A good rule makes exactly one number odd. After you find your answer, quickly test one more property (prime, parity, digit sum). If another property points to a different number and looks equally natural, re-read the options. In a well-made question every natural rule points to the same answer.

Common traps

Treating 1 as prime (it is not) or 2 as composite (2 is the only even prime).
Forgetting that a number can be both a square and a cube (64 = 8² = 4³, 729 = 27² = 9³).
Stopping at "all are odd" or "all are even" - parity is almost never the whole rule when all four match.

Quick revision

Test	Fast check
Prime	Divide by 2, 3, 5, 7, 11, 13
Square	Last digit not 2, 3, 7, 8; know 1²-30²
Cube	Know 1³-12³
Multiple of 3 / 9	Digit sum
Multiple of 11	Alternate-digit difference 0 or 11
Digit rule	Same digit sum / product
Two-level	Look at roots, then parity
Forms	$n^2 \pm 1$ , $n^3 \pm 1$ , $n(n+1)$

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Prime vs compositevery common3 practice Q

How to spot it:

Four numbers, mostly odd, that 'look' prime. One is a hidden product like 91 or 143 (or one is prime among composites).

p \text{ prime} \iff \text{no divisor } 2 \le d \le \sqrt{p}

Divide each number by 2, 3, 5, 7, 11, 13.
Mark which ones have a factor.
The single number on the other side is the answer.

Why it works: primes are the setter's favourite group; hidden composites (91, 119, 143) are the trap.

Example: Find the odd one: 61, 67, 71, 77.

61, 67, 71 are prime. 77 = 7 × 11 is composite. Answer: 77.

Type 2: Perfect square / perfect cubevery common2 practice Q

How to spot it:

Numbers such as 169, 196, 225 or 216, 343, 512 - familiar powers with one stranger.

n^2,\; n^3

Recall the square/cube table.
Mark which numbers are exact squares or cubes.
The one that is not is odd.

Why it works: a number between two known powers cannot be a power.

Example: Find the odd one: 169, 196, 225, 250.

169 = 13², 196 = 14², 225 = 15². 250 lies between 15² and 16² = 256. Answer: 250.

Type 3: Multiples / divisibilitycommon2 practice Q

How to spot it:

Three numbers are multiples of one number (7, 9, 11, 13...), often shown by a digit rule.

11 \mid n \iff (\text{odd-place sum}) - (\text{even-place sum}) \equiv 0 \pmod{11}

Find the common divisor of most of the numbers (try 7, 9, 11, 13).
Use divisibility rules for big numbers (digit sum for 9, alternate sum for 11).
The number that is not a multiple is odd.

Why it works: the setter picks a divisor, writes three multiples and one near-miss.

Example: Find the odd one: 39, 65, 91, 111.

39 = 13 × 3, 65 = 13 × 5, 91 = 13 × 7. 111 = 3 × 37 is not a multiple of 13. Answer: 111.

Type 4: Digit sum / digit productcommon2 practice Q

How to spot it:

Numbers with no clear prime/square pattern, often three-digit, whose digits look shuffled.

S(n) = \sum \text{digits},\; P(n) = \prod \text{digits}

Add the digits of each number.
If sums differ, multiply the digits.
The number whose sum or product differs is odd.

Why it works: digits can be rearranged freely while keeping the same sum and product.

Example: Find the odd one: 345, 453, 534, 546.

Digit sums: 12, 12, 12, 15. Answer: 546.

Type 5: Two-level: same power, odd/even rootcommon2 practice Q

How to spot it:

All four are squares (or all cubes), so the first rule does not decide.

Write the root of each number.
Check roots for odd/even or prime/composite.
The number whose root breaks the pattern is odd.

Why it works: the first property is shared by all four to trap you; the answer hides in the roots.

Example: Find the odd one: 27, 125, 343, 512.

All cubes: 3³, 5³, 7³, 8³. Roots 3, 5, 7 are odd; 8 is even. Answer: 512.

Type 6: Number forms (n²±1, n³±1, n(n+1))occasional3 practice Q

How to spot it:

Numbers sit just beside squares or cubes (50, 65, 82 or 63, 124, 215), or are products of two consecutive numbers.

n^2+1,\; n^3-1,\; n(n+1)

Check each number against the nearest square and cube.
Write it as $n^2 \pm 1$ , $n^3 \pm 1$ or $n(n+1)$ .
The number that does not fit the common form is odd.

Why it works: the setter builds three numbers from one formula and changes the sign or the base for the fourth.

Example: Find the odd one: 50, 65, 82, 99.

50 = 7² + 1, 65 = 8² + 1, 82 = 9² + 1. 99 = 10² − 1. Answer: 99.

Formulas

Divisibility by 11

|S_{odd} - S_{even}| \equiv 0 \pmod{11}

S = sums of alternate digits; e.g. 2728: (2+2)-(7+8) = -11 → divisible.

Perfect square endings

n^2 \in \{0,1,4,5,6,9\}

A square never ends in 2, 3, 7 or 8 — instant elimination.

Digit sum 9 rule

9 \mid n \iff S(n) \equiv 0 \pmod 9

Numbers with digit sum exactly 9: 45, 54, 63, 72, 81.

Shortcut tricks

⚡ Last-digit scan for squares

Squares end only in 0, 1, 4, 5, 6, 9. If three numbers are squares, any option ending in 2/3/7/8 is instantly odd — no squaring needed.

Example: Q. 144, 324, 576, 500

Sol. All end in even digits, but squares must end in 0/1/4/5/6/9 — 500 ends in 0 (possible), so test values: 144 = 12², 324 = 18², 576 = 24², 500 is between 22² = 484 and 23² = 529 → 500 is not a square.

Ending check eliminates; root check confirms — together under 10 seconds.

⚡ Prime check by small divisors

To test primality divide by 2, 3, 5, 7 only (for numbers under 121). 33 = 3 × 11, 51 = 3 × 17, 57 = 3 × 19, 91 = 7 × 13 — CGL's favourite fake primes.

Example: Q. 17, 51, 41, 43

Sol. 17, 41, 43 are prime; 51 = 3 × 17 → 51.

Memorise the fake-prime list: 33, 51, 57, 91, 119, 133, 143 (11×13).

Where students lose marks

Ignoring that 1 is neither prime nor composite; 2 is the only even prime.
Calling 56 odd among multiples of 7 without noticing 56 is also a multiple — check the rule against every option.
Overlooking a two-level structure (odd squares vs even squares) and marking a valid square as the answer for the wrong reason.

Practice sets — 20 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 10 questions

Suggested time 5 min · wrong answers go to your mistake notebook automatically.