Question 1

Find the odd number out in the sequence: 3, 5, 11, 14, 17, 21.

Accepted Answer

Problem restatement Identify the single term that does not follow the same basic property as the others in the list 3, 5, 11, 14, 17, 21. Given data Sequence: 3, 5, 11, 14, 17, 21 Task: Find the one number that differs in a simple, common property. Concept / Approach Start with the simplest structural checks: parity (odd/even), perfect squares/cubes, small prime/composite patterns, and quick divisibility. The simplest clear separator here is parity. Step-by-step check 3 → odd 5 → odd 11 → odd 14 → even 17 → odd 21 → odd Exactly one term (14) is even; all others are odd. Verification / Alternative Prime/composite is not decisive here (21 and 14 are composite, others are prime), but parity produces a unique outlier: only 14 is even. Common pitfalls Overfitting: trying to impose a complex pattern when a basic property (odd/even) already yields a unique answer. Mixing prime vs composite leads to two composites (14 and 21), not a single outlier. Final Answer 14

Question 2

Find the odd number out in the sequence: 8, 27, 64, 100, 125, 216, 343.

Accepted Answer

Problem restatement Among 8, 27, 64, 100, 125, 216, 343, pick the single term that does not share the same core property. Given data Sequence: 8, 27, 64, 100, 125, 216, 343 Task: Find the unique outlier. Concept / Approach Check for perfect powers. These look like cubes around small integers. Step-by-step validation 8 = 23 (cube) 27 = 33 (cube) 64 = 43 (cube) 100 ≠ n3 for integer n (not a cube) 125 = 53 (cube) 216 = 63 (cube) 343 = 73 (cube) Every term is a perfect cube except 100. Verification / Alternative Nearest cubes around 100 are 43 = 64 and 53 = 125; 100 lies between them and is not a cube. Common pitfalls Seeing 100 as a perfect square (102) and assuming all are squares—several terms are not squares but are cubes. Final Answer 100

Question 3

Find the odd number out in the sequence: 10, 25, 45, 54, 60, 75, 80.

Accepted Answer

Problem restatement From 10, 25, 45, 54, 60, 75, 80, identify the single outlier. Given data Sequence: 10, 25, 45, 54, 60, 75, 80 Concept / Approach Check divisibility patterns (by 2, 3, 5). The presence of many multiples of 5 suggests a quick screen. Step-by-step validation 10 → divisible by 5 25 → divisible by 5 45 → divisible by 5 54 → not divisible by 5 60 → divisible by 5 75 → divisible by 5 80 → divisible by 5 All except 54 are multiples of 5; hence 54 is the outlier. Verification / Alternative Divisibility by 5 requires the last digit 0 or 5. Only 54 ends with neither. Common pitfalls Checking only evenness—several terms are even, but that does not isolate a unique outlier. Final Answer 54

Question 4

Find the odd number out in the sequence: 396, 462, 572, 427, 671, 264.

Accepted Answer

Problem restatement Among 396, 462, 572, 427, 671, 264, select the number that does not share the principal divisibility property of the rest. Given data Sequence: 396, 462, 572, 427, 671, 264 Concept / Approach Test for a strong common factor. A quick check shows many are divisible by 11. Use the 11-divisibility rule or direct division. Step-by-step validation 396 ÷ 11 = 36 exactly (divisible by 11) 462 ÷ 11 = 42 exactly (divisible by 11) 572 ÷ 11 = 52 exactly (divisible by 11) 427 ÷ 11 = 38 remainder 9 (not divisible by 11) 671 ÷ 11 = 61 exactly (divisible by 11) 264 ÷ 11 = 24 exactly (divisible by 11) All listed numbers are multiples of 11 except 427. Verification / Alternative 11-rule: Alternating sum of digits is a multiple of 11. 396: (3 − 9 + 6) = 0 → divisible 462: (4 − 6 + 2) = 0 → divisible 572: (5 − 7 + 2) = 0 → divisible 427: (4 − 2 + 7) = 9 → not divisible 671: (6 − 7 + 1) = 0 → divisible 264: (2 − 6 + 4) = 0 → divisible Common pitfalls Using only divisibility by 3: that splits the set but does not produce a unique outlier. Final Answer 427

Question 5

Find the odd number out in the sequence: 6, 9, 15, 21, 24, 28, 30.

Accepted Answer

Problem restatement From 6, 9, 15, 21, 24, 28, 30, determine the single term that does not share the common divisibility property. Given data Sequence: 6, 9, 15, 21, 24, 28, 30 Concept / Approach Most terms look like multiples of 3. Use the sum-of-digits test for divisibility by 3. Step-by-step validation 6 → 6 (sum 6) → divisible by 3 9 → 9 (sum 9) → divisible by 3 15 → 1+5=6 → divisible by 3 21 → 2+1=3 → divisible by 3 24 → 2+4=6 → divisible by 3 28 → 2+8=10 → not divisible by 3 30 → 3+0=3 → divisible by 3 Exactly one term (28) is not divisible by 3. Verification / Alternative Direct division confirms: 28 ÷ 3 = 9 remainder 1 (not a multiple of 3). Common pitfalls Relying on evenness: several terms are even, so parity does not isolate a unique outlier. Final Answer 28

Question 6

Find the odd number out in the sequence: 1, 4, 9, 16, 23, 25, 36.

Accepted Answer

Problem restatement Among 1, 4, 9, 16, 23, 25, 36, identify the unique term that does not share the core property of the others. Given data Sequence: 1, 4, 9, 16, 23, 25, 36 Concept / Approach These are mostly perfect squares of consecutive integers. Step-by-step validation 1 = 12 4 = 22 9 = 32 16 = 42 23 ≠ n2 for integer n 25 = 52 36 = 62 All are perfect squares except 23. Verification / Alternative Note that 23 lies strictly between 42=16 and 52=25, so it cannot be a perfect square. Common pitfalls Misreading 23 as 62 (which is 36), or assuming it fits the square pattern. Final Answer 23

Question 7

Find the odd number out in the sequence: 1, 4, 9, 16, 20, 36, 49.

Accepted Answer

Problem restatement From 1, 4, 9, 16, 20, 36, 49, choose the number that does not share the main property with the rest. Given data Sequence: 1, 4, 9, 16, 20, 36, 49 Concept / Approach The terms are largely perfect squares of small integers; check each. Step-by-step validation 1 = 12 4 = 22 9 = 32 16 = 42 20 ≠ n2 for integer n 36 = 62 49 = 72 All are perfect squares except 20. Verification / Alternative 20 lies between 42=16 and 52=25, confirming it is not a perfect square. Common pitfalls Assuming every term is a square by pattern recognition without checking 20 individually. Final Answer 20

Question 8

Sequence given without instruction: '2, 5, 10, 17, 26, 37, 50, 64' (task not specified).

Accepted Answer