Question 1

Find the odd term out: 3, 7, 15, 39, 63, 127, 255, 511 Exactly one number breaks the intended “2^n − 1” pattern.

Accepted Answer

Introduction / Context:Many fast-growing series follow the form 2^n − 1: 3, 7, 15, 31, 63, 127, 255, 511, … Spotting any deviation from this canonical list identifies the odd man out. Given Data / Assumptions: Presented sequence: 3, 7, 15, 39, 63, 127, 255, 511. Known reference pattern: 3, 7, 15, 31, 63, 127, 255, 511. Exactly one number is inconsistent. Concept / Approach:Compare term-by-term with 2^n − 1 for n = 2 through 9. The mismatch pinpoints the erroneous entry. Step-by-Step Solution:2^2 − 1 = 3 ✓2^3 − 1 = 7 ✓2^4 − 1 = 15 ✓2^5 − 1 = 31 (but the series shows 39 ✗)2^6 − 1 = 63 ✓2^7 − 1 = 127 ✓2^8 − 1 = 255 ✓2^9 − 1 = 511 ✓ Verification / Alternative check:Replacing 39 with 31 yields a perfect 2^n − 1 run. No other power-based family fits all remaining values so cleanly. Why Other Options Are Wrong: 7 / 15 / 63 belong exactly to 2^3 − 1, 2^4 − 1, 2^6 − 1 respectively and are consistent. Common Pitfalls: Assuming a multiplicative rule like “×2 + 1” inconsistently; the strict 2^n − 1 check is definitive. Final Answer:39

Question 2

Identify the wrong number in the division-by-4 sequence: 40960, 10240, 2560, 640, 200, 40, 10

Accepted Answer

Introduction / Context:When each term equals the previous term divided by a constant factor, one erroneous division immediately breaks the rhythm. Here, division by 4 appears to be the governing rule. Given Data / Assumptions: Sequence: 40960, 10240, 2560, 640, 200, 40, 10. Hypothesis: each term = previous term / 4. Concept / Approach:Check each ratio. A single ratio that is not 1/4 highlights the wrong term. If a wrong intermediate is corrected, the remainder should fall into place automatically. Step-by-Step Solution:10240 = 40960 / 4 (OK)2560 = 10240 / 4 (OK)640 = 2560 / 4 (OK)Next should be 640 / 4 = 160 (but given: 200 ✗)40 = 160 / 4 ⇒ would be correct after fixing 16010 = 40 / 4 ⇒ consistent Verification / Alternative check:With 160 in place of 200, the series becomes a perfect divide-by-4 chain. No other constant factor explains the earlier steps so precisely. Why Other Options Are Wrong: 640 / 2560 / 40 are each exactly one quarter of the prior term; they follow the rule. Common Pitfalls: Treating the middle anomaly as a pattern change; in clean constant-ratio series, a single error is typical. Final Answer:200

Question 3

Odd term in a factorial-like progression: 1, 1, 2, 6, 24, 96, 720 Exactly one number is inconsistent with n! progression.

Accepted Answer

Introduction / Context:Recognizing factorials is a staple skill in series: 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 6! = 720. A single non-factorial value stands out as the odd term. Given Data / Assumptions: Given: 1, 1, 2, 6, 24, 96, 720. True factorials up to 6!: 1, 1, 2, 6, 24, 120, 720. Concept / Approach:Match each term to n! for n = 0 through 6. The mismatch is the wrong number. Factorials grow rapidly; any off-by-one multipliers become obvious. Step-by-Step Solution:1 = 0! (or 1!) ✓1 = 1! ✓2 = 2! ✓6 = 3! ✓24 = 4! ✓96 ≠ 5! (should be 120) ✗720 = 6! ✓ Verification / Alternative check:Since 96 = 24 * 4, while 5! must be 24 * 5, replacing 96 by 120 restores the perfect factorial sequence. Why Other Options Are Wrong: 6 / 24 / 720 are exactly 3!, 4!, 6! and thus correct. Common Pitfalls: Confusing powers (e.g., 4^3 = 64) with factorials; here only 96 is off. Final Answer:96

Question 4

Find the wrong number in the series: 5, 15, 30, 135, 405, 1215, 3645

Accepted Answer

Introduction / Context:This series is intended to be a clean “multiply by 3” chain. One term violates the constant ratio, and catching that restores the simple geometric progression. Given Data / Assumptions: Given: 5, 15, 30, 135, 405, 1215, 3645. Ideal 3× progression: 5, 15, 45, 135, 405, 1215, 3645. Concept / Approach:Compute ratios term by term. A constant ratio of 3 should connect successive terms. Any deviation indicates the wrong entry. Step-by-Step Solution:15 / 5 = 3 (OK)30 / 15 = 2 (should have been 3) ✗135 / 45 = 3 (if 45 had been present)405 / 135 = 31215 / 405 = 33645 / 1215 = 3 Verification / Alternative check:Replacing 30 with 45 yields a perfect geometric sequence with ratio 3. All later terms already adhere to the 3× rule. Why Other Options Are Wrong: 405 / 1215 / 3645 are consistent 3× successors; removing any would break the clean ratio. Common Pitfalls: Overlooking a single early error and trying to fit mixed ratios thereafter. The simplest constant ratio is preferable. Final Answer:30

Question 5

Spot the odd term in a “+ odd numbers” pattern: 2, 5, 10, 17, 26, 37, 50, 64

Accepted Answer

Introduction / Context:This sequence adds consecutive odd numbers to generate each next term. Identifying a single break in that steadily growing odd-difference pattern reveals the odd term. Given Data / Assumptions: Series: 2, 5, 10, 17, 26, 37, 50, 64. Target difference pattern: +3, +5, +7, +9, +11, +13, +15, … Concept / Approach:Compute differences and compare to the intended odd sequence. The term that fails to reflect the correct next odd difference is the odd one out. Step-by-Step Solution:5 − 2 = 310 − 5 = 517 − 10 = 726 − 17 = 937 − 26 = 1150 − 37 = 13Next should be +15 ⇒ 50 + 15 = 65 (but given: 64 ✗) Verification / Alternative check:Replacing 64 with 65 reestablishes the clean +3, +5, +7, +9, +11, +13, +15 progression. All earlier steps already match this rule perfectly. Why Other Options Are Wrong: 50 / 26 / 37 are fully consistent with consecutive odd increments; only the last step is off by 1. Common Pitfalls: Treating small end-of-series typos as pattern changes; here the odd-number ladder is definitive. Final Answer:64

Question 6

Find the odd number out: 331, 482, 551, 263, 383, 242, 111

Accepted Answer

Introduction / Context:Some odd-man-out puzzles rely on a distinctive arithmetic property rather than a recursive rule. Here, digit patterns and basic divisibility tests are good first checks to isolate a unique number among peers. Given Data / Assumptions: Numbers: 331, 482, 551, 263, 383, 242, 111. We are to find the single term that differs by a clear numeric property from the rest. Concept / Approach:Apply divisibility-by-3 test: a number divisible by 3 must have digit-sum divisible by 3. Evaluate each candidate and see whether only one satisfies this condition, making it the “odd man out.” Step-by-Step Solution:331 → digit sum 3+3+1 = 7 (not divisible by 3)482 → 4+8+2 = 14 (not divisible by 3)551 → 5+5+1 = 11 (not divisible by 3)263 → 2+6+3 = 11 (not divisible by 3)383 → 3+8+3 = 14 (not divisible by 3)242 → 2+4+2 = 8 (not divisible by 3)111 → 1+1+1 = 3 (divisible by 3) Verification / Alternative check:Beyond the divisibility test, 111 also uniquely comprises three identical digits, a secondary distinctive property. No other number in the set shares both features, reinforcing the choice. Why Other Options Are Wrong: 263 / 383 / 242: All fail the divisibility-by-3 test and lack the “all digits equal” feature; they do not stand out uniquely. Common Pitfalls: Forcing a complicated recurrence; in some odd-man-out items, a single clean property (like divisibility by 3) is intended. Final Answer:111

Question 7

Identify the wrong term in a square-number increment pattern: 1, 2, 6, 15, 31, 56, 91

Accepted Answer

Introduction / Context:This series is formed by adding consecutive perfect squares to build each next term. A single off-by-one near the end reveals the incorrect entry. Given Data / Assumptions: Series: 1, 2, 6, 15, 31, 56, 91. Intended increments: +1^2, +2^2, +3^2, +4^2, +5^2, +6^2. Concept / Approach:Compute successive differences and see whether they match 1, 4, 9, 16, 25, 36. The first five steps align; the last does not, exposing the wrong term. Step-by-Step Solution:2 − 1 = 1 (1^2)6 − 2 = 4 (2^2)15 − 6 = 9 (3^2)31 − 15 = 16 (4^2)56 − 31 = 25 (5^2)Next should be +36 ⇒ 56 + 36 = 92 (but given: 91) Verification / Alternative check:Correcting 91 to 92 reestablishes the perfect-squares increment ladder. No other simple pattern explains all earlier steps as cleanly. Why Other Options Are Wrong: 31 / 56 / 15 each sits exactly at the right cumulative square-sum. Common Pitfalls: Misreading late-series arithmetic by 1; always check the exact square increment expected. Final Answer:91

Question 8

Odd one out among squares: 25, 36, 49, 81, 121, 169, 225

Accepted Answer

Introduction / Context:All listed numbers are perfect squares, but there is a deeper structural contrast: one is the square of an even integer while the rest are squares of odd integers. Identifying that unique parity distinguishes the odd term. Given Data / Assumptions: Squares listed: 25 (5^2), 36 (6^2), 49 (7^2), 81 (9^2), 121 (11^2), 169 (13^2), 225 (15^2). Only one is an even square. Concept / Approach:Classify each term by the parity of its square root. If exactly one is even and all others are odd, that even square is the odd man out. Step-by-Step Solution:25 = 5^2 (odd)36 = 6^2 (even) ← unique49 = 7^2 (odd)81 = 9^2 (odd)121 = 11^2 (odd)169 = 13^2 (odd)225 = 15^2 (odd) Verification / Alternative check:Other properties (e.g., prime vs composite roots) do not segregate uniquely. Parity does: only 36 arises from an even root, 6. Why Other Options Are Wrong: 49 / 121 / 169: all are odd squares like the majority of the list. Common Pitfalls: Searching for complex relations when a simple structural difference (even vs odd square) suffices. Final Answer:36

Question 9

Find the wrong term in steadily widening even jumps: 56, 72, 90, 110, 132, 150

Accepted Answer

Introduction / Context:This series increases by even differences that themselves rise by 2 at each step: +16, +18, +20, +22, +24, … A single terminal value breaks this predictable pace. Given Data / Assumptions: Sequence: 56, 72, 90, 110, 132, 150. Observed differences: 16, 18, 20, 22, (should be 24). Concept / Approach:List differences. If the pattern is “add 2 to the difference each step,” then after +22 the next must be +24. Apply it to confirm the correct terminal value. Step-by-Step Solution:72 − 56 = 1690 − 72 = 18110 − 90 = 20132 − 110 = 22Expected next: 132 + 24 = 156 (but given 150) Verification / Alternative check:Replacing 150 with 156 restores the uniform “+2 to the difference” ladder of even numbers. Any other replacement distorts at least one earlier step. Why Other Options Are Wrong: 72 / 110 / 132 all adhere to the intended differences; only the last step is low by 6. Common Pitfalls: Mistaking a steady second-difference pattern; always examine the sequence of differences themselves. Final Answer:150

Question 10

Locate the odd term in alternating increments: 6, 13, 18, 25, 30, 37, 40

Accepted Answer

Introduction / Context:Alternating-difference series commonly repeat a two-step cycle (e.g., +7, +5, +7, +5, …). One term can violate the cycle, and the break is exposed by inspecting consecutive differences. Given Data / Assumptions: Sequence: 6, 13, 18, 25, 30, 37, 40. Hypothesized cycle: +7, +5, +7, +5, +7, +5, … Concept / Approach:Compute differences to confirm the cycle. If a final step fails to follow the expected +5 after a +7, the last number is suspect. Step-by-Step Solution:13 − 6 = +718 − 13 = +525 − 18 = +730 − 25 = +537 − 30 = +7Next should be +5 ⇒ 37 + 5 = 42 (but given 40) Verification / Alternative check:Substituting 42 for 40 reinstates the strict +7, +5 alternation. Earlier steps already fit perfectly, confirming the intended pattern. Why Other Options Are Wrong: 25 / 30 / 37 align with the alternating increments; they are not the anomaly. Common Pitfalls: Smoothing over a two-step cycle with an average increment; the alternation must be respected. Final Answer:40

Question 11

Odd term in a descending-division pattern: 46080, 3840, 384, 48, 24, 2, 1

Accepted Answer

Introduction / Context:Some series apply successive divisions by decreasing even numbers (e.g., ÷12, ÷10, ÷8, ÷6, ÷4, ÷2). A single mis-division yields a standout term. Here we test that descending-divisor hypothesis. Given Data / Assumptions: Sequence: 46080, 3840, 384, 48, 24, 2, 1. Suspected divisors: 12, 10, 8, 6, 4, 2 in order. Concept / Approach:Check each step against the intended divisor chain. Where the quotient disagrees, the respective term is wrong. After correction, later terms should follow smoothly. Step-by-Step Solution:46080 ÷ 12 = 3840 ✓3840 ÷ 10 = 384 ✓384 ÷ 8 = 48 ✓Next should be 48 ÷ 6 = 8 (but given: 24 ✗)Then 8 ÷ 4 = 2, and 2 ÷ 2 = 1 ✓ Verification / Alternative check:Once 24 is replaced by 8, the chain becomes perfectly coherent with divisors 12, 10, 8, 6, 4, 2. The final terms (2, 1) already match the intended finishing steps. Why Other Options Are Wrong: 384 / 2 / 1 all fit exactly with the nominated divisors. They are not the anomalies. Common Pitfalls: Assuming constant ratios; here the ratio changes in a controlled descending manner. Final Answer:24

Question 12

Find the wrong number in an alt ×1.5 and ÷3 pattern: 36, 54, 18, 27, 9, 18.5, 4.5

Accepted Answer

Introduction / Context:This alternating series multiplies by 1.5 (i.e., ×3/2) and then divides by 3, repeating the two-step cycle. One term fails to obey the cycle and becomes the odd man out. Given Data / Assumptions: Sequence: 36, 54, 18, 27, 9, 18.5, 4.5. Hypothesized cycle: ×1.5, ÷3, ×1.5, ÷3, … Concept / Approach:Apply the cycle step-by-step. Any value that does not match the computed result under the cycle is the erroneous term. Step-by-Step Solution:36 × 1.5 = 54 ✓54 ÷ 3 = 18 ✓18 × 1.5 = 27 ✓27 ÷ 3 = 9 ✓9 × 1.5 = 13.5 (but given: 18.5 ✗)13.5 ÷ 3 = 4.5 (final term is consistent if the prior were corrected) Verification / Alternative check:Correcting 18.5 to 13.5 produces the perfectly alternating sequence: 36, 54, 18, 27, 9, 13.5, 4.5. Why Other Options Are Wrong: 54 / 18 / 4.5 each results directly from the stipulated ×1.5 or ÷3 step. Common Pitfalls: Mistaking an average multiplier; ensure alternation is respected (multiply then divide). Final Answer:18.5

Question 13

Odd term in an “add 1,2,3,4 then 5,6,7,8 …” ladder: 4, 5, 7, 10, 14, 18, 25, 32

Accepted Answer

Introduction / Context:This series uses a growing ladder of consecutive additions grouped in runs: +1, +2, +3, +4, then +5, +6, +7, +8, and so on. A single term undercuts the expected addition, revealing itself as the odd one out. Given Data / Assumptions: Sequence: 4, 5, 7, 10, 14, 18, 25, 32. Hypothesized additions: +1, +2, +3, +4, +5, +6, +7, +8. Concept / Approach:Compute successive differences and match them against the intended ladder. The incorrect step will be short by the expected increment. Step-by-Step Solution:5 − 4 = +17 − 5 = +210 − 7 = +314 − 10 = +418 − 14 = +4 (should be +5) ✗25 − 18 = +7 (as if the ladder resumed later)32 − 25 = +7 (should be +8 if continuing strictly) Verification / Alternative check:If 18 were 19 (adding +5 correctly), the next should be +6 to 25, then +7 to 32, and +8 would follow. The earliest break is at 18, making it the odd term. Why Other Options Are Wrong: 7 / 14 / 32 fit or are reconcilable once the +5 step is restored; 18 uniquely under-delivers by 1. Common Pitfalls: Overlooking grouped runs of consecutive additions; ensuring the ladder increments correctly pinpoints the anomaly. Final Answer:18

Question 14

Odd one out in a digit-property set Identify the number that does not follow the common digit rule in the set: 253, 136, 352, 460, 324, 631, 244.

Accepted Answer

Introduction / Context:This odd-man-out puzzle asks you to detect a hidden pattern across multi-digit numbers and then pick the single value that breaks that rule. Many such questions rely on simple arithmetic relations among digits (sum, difference, parity, or primality). Here, checking divisibility and compositeness often reveals the correct outlier quickly and reliably. Given Data / Assumptions: Numbers to test: 253, 136, 352, 460, 324, 631, 244. We assume the same base-10 digit interpretation for all terms. We look for a common numeric property present in most items but violated by exactly one item. Concept / Approach:For digit-based sets where no clear arithmetic progression exists, a fast screen is to test primality or obvious factorization. If six items share “composite” status and a single one is prime, the prime is the odd one out. This method is robust when digit-relationship hypotheses (like middle digit equals sum of outer digits) do not hold uniformly across the set. Step-by-Step Solution:Check 253 = 11 * 23 ⇒ composite.Check 136 = 8 * 17 ⇒ composite.Check 352 = 32 * 11 ⇒ composite.Check 460 = 46 * 10 ⇒ composite.Check 324 = 18 * 18 ⇒ composite.Check 244 = 4 * 61 ⇒ composite.Check 631: test small primes (2,3,5,7,11,13,17,19,23). Not divisible by 2 or 5. Sum of digits = 10 ⇒ not divisible by 3. 631/7 ≠ integer; 631/11 ≠ integer; 631/13 ≈ 48.5; 631/17 ≈ 37.1; 631/19 ≈ 33.2; 631/23 ≈ 27.4. No small factor found; 631 is prime. Verification / Alternative check:Since every other number factors cleanly, and 631 resists all standard small-prime divisibility tests, 631 is the only prime in the set. That makes it the clear outlier by the “composite vs. prime” property. Why Other Options Are Wrong:244, 352, 136, and 460 are all composite (each has nontrivial factors), so they fit the dominant pattern and cannot be the odd one out. Common Pitfalls:Getting distracted by digit-sum tricks (e.g., “first + last = middle”) that do not hold consistently, or assuming a hidden sequence when the task is a classification by number type. Always test simple number-theory properties early for odd-man-out items. Final Answer:631

Question 15

Prime number sequence continuation Given the primes: 11, 13, 17, 19, 23, 29, 31, 37, 41, what is the next prime?

Accepted Answer

Introduction / Context:This is a straightforward prime sequence continuation problem. You are given a strictly increasing list of prime numbers and asked to identify the next prime. Recognizing primes and skipping composites is the key skill tested here. Given Data / Assumptions: Sequence shown: 11, 13, 17, 19, 23, 29, 31, 37, 41. Task: choose the next prime greater than 41. We assume standard definition of prime: an integer greater than 1 with no positive divisors other than 1 and itself. Concept / Approach:List consecutive integers after 41 and test primality: 42, 43, 44, 45, 46, 47, etc. Eliminate even numbers and those divisible by small primes (3, 5). The first candidate that passes basic divisibility tests is the next prime. Step-by-Step Solution:42 is even ⇒ composite.43: test divisibility by primes ≤ √43 (2, 3, 5). Not divisible ⇒ 43 is prime.Thus, the next prime after 41 is 43. Verification / Alternative check:Continue checking: 44 is even; 45 divisible by 3 and 5; 46 even; 47 is also prime but it comes after 43. The immediate successor prime to 41 is indeed 43. Why Other Options Are Wrong:47 and 53 are primes but not the immediate next; 51 and 49 are composite (51 = 3 * 17, 49 = 7 * 7). Common Pitfalls:Jumping to a larger prime (like 47) without checking the smaller candidate 43, or forgetting that “next” means the first prime greater than 41, not any future prime. Final Answer:43

Question 16

Alternate operations series (+3 then ×2) Complete the pattern: 5, 10, 13, 26, 29, 58, 61, ( ... ).

Accepted Answer

Introduction / Context:Many numeric series alternate between two simple operations. Detecting the alternation and applying it to the last term yields the next value quickly. This question mixes addition and multiplication in a predictable cycle. Given Data / Assumptions: Series: 5, 10, 13, 26, 29, 58, 61, ... We expect a repeating two-step operation. Concept / Approach:Check adjacent jumps: 5→10 (+5), 10→13 (+3), 13→26 (×2), 26→29 (+3), 29→58 (×2), 58→61 (+3). After +3 typically follows ×2. Therefore, apply ×2 to 61 to get the next term. Step-by-Step Solution:5 → 10: ×2.10 → 13: +3.13 → 26: ×2.26 → 29: +3.29 → 58: ×2.58 → 61: +3.61 → next: ×2 ⇒ 122. Verification / Alternative check:Write the pattern explicitly as (×2, +3) repeating. Starting from 5 gives the exact sequence given; so the next term must be 122. Why Other Options Are Wrong:125, 128, 64, 120 do not result from applying ×2 to 61 in this alternating pattern; they correspond to other operations not present here. Common Pitfalls:Assuming a constant difference or mixing the order (+3 then ×2 vs. ×2 then +3). Always trace from term to term to lock the alternation. Final Answer:122

Question 17

Prime cubes pattern — identify the odd one out Numbers: 8, 27, 125, 343, 1331. Which option is the odd one out?

Accepted Answer

Introduction / Context:Odd-man-out questions sometimes contain a perfectly consistent set, expecting you to recognize that no single element breaks the rule. Each given number should be examined for a unifying property (such as being a power of a prime). Given Data / Assumptions: List: 8, 27, 125, 343, 1331. Task: pick the odd one out from the options, or conclude none of the listed numbers is odd relative to the others. Concept / Approach:Check whether each term can be written as p^3 (a cube of a prime p). If all satisfy that, then the set is uniform and “None of these” (no odd element) is correct. Step-by-Step Solution:8 = 2^3 (prime base 2).27 = 3^3 (prime base 3).125 = 5^3 (prime base 5).343 = 7^3 (prime base 7).1331 = 11^3 (prime base 11).All are cubes of consecutive primes (2, 3, 5, 7, 11). Verification / Alternative check:No term deviates from the “cube of a prime” rule. Hence the set is consistent, and there is no odd one out among the numbers themselves. Why Other Options Are Wrong:Selecting 1331, 343, 125, or 27 would be incorrect because each fits the same rule as the rest. The correct choice is that none of these individual numbers stands out as different. Common Pitfalls:Overthinking and trying to invent a difference when a clean, uniform pattern exists. Remember that “None of these” is often the right response when all items share the same defining property. Final Answer:None of these

Question 18

Alternating pattern (+4, ×2): find the odd one out Sequence shown: 10, 14, 28, 32, 64, 68, 132. Which term breaks the pattern?

Accepted Answer

Introduction / Context:This is a classic alternating-operations series. When two operations repeat (for example, +a then ×b), any deviation from the expected outcome signals the odd term. Your goal is to spot the expected value and compare it with the one given. Given Data / Assumptions: Series: 10, 14, 28, 32, 64, 68, 132. Hypothesis: “+4, then ×2” repeating. Concept / Approach:Apply the supposed pattern step by step and see where the provided term disagrees. If a computed value differs from the listed one, that listed value is the odd-man-out. Step-by-Step Solution:10 → 14: +4 (fits).14 → 28: ×2 (fits).28 → 32: +4 (fits).32 → 64: ×2 (fits).64 → 68: +4 (fits).68 → expected next: ×2 ⇒ 136, but listed is 132.Therefore, 132 is the odd term. Verification / Alternative check:Recomputing several steps confirms the consistency of the rule and isolates only the last step as inconsistent. That makes 132 the unique outlier. Why Other Options Are Wrong:32, 68, 28, and 64 all agree with the alternating “+4, ×2” pattern at their positions. They cannot be the odd ones out. Common Pitfalls:Confusing the operation order (×2 then +4 instead of +4 then ×2), or trying to force a different rule mid-sequence. Always verify consistency through multiple consecutive steps. Final Answer:132

Question 19

Product pattern n*(2n+1): find the odd one out Numbers: 3, 10, 21, 36, 55, 70, 105. Which number does not fit n*(2n+1)?

Accepted Answer

Introduction / Context:Some sequences follow the form n*(2n+1) for consecutive integers n. Spotting this form helps check whether each number belongs. If exactly one number breaks the identity, that term is the odd one out. Given Data / Assumptions: Candidates: 3, 10, 21, 36, 55, 70, 105. Proposed form: n*(2n+1) for integer n ≥ 1. Concept / Approach:Compute n*(2n+1) for n = 1, 2, 3, ... and compare with the terms. If one term cannot be expressed this way for any integer n while neighbors can, it is the odd one out. Step-by-Step Solution:n=1 ⇒ 1*(3)=3 ✔n=2 ⇒ 2*(5)=10 ✔n=3 ⇒ 3*(7)=21 ✔n=4 ⇒ 4*(9)=36 ✔n=5 ⇒ 5*(11)=55 ✔n=6 ⇒ 6*(13)=78 (not 70) ✘n=7 ⇒ 7*(15)=105 ✔Thus 70 does not fit the pattern; expected value between 55 and 105 would have been 78. Verification / Alternative check:Check differences: 3→10 (+7), 10→21 (+11), 21→36 (+15), 36→55 (+19), 55→78 (+23 expected), then 78→105 (+27). The +23 step is missing in the list and 70 appears instead, confirming the inconsistency. Why Other Options Are Wrong:105, 36, 55, and 21 each exactly match n*(2n+1) for n = 7, 4, 5, and 3 respectively. They fit the rule. Common Pitfalls:Assuming a constant difference or trying to fit a linear rule when the quadratic-like growth suggests a formula such as n*(2n+1). Final Answer:70

Question 20

Composite vs. prime classification — pick the odd one out Set: 4, 6, 8, 9, 10, 11, 12. Which single number is different from the rest?

Accepted Answer

Introduction / Context:Odd-man-out questions often hinge on a basic number-theory property: most elements share “composite” status while one is “prime.” Identifying primality quickly separates the outlier from the rest. Given Data / Assumptions: Numbers provided: 4, 6, 8, 9, 10, 11, 12. We use the standard definition of prime and composite. Concept / Approach:Test each number: composite numbers have nontrivial factors; primes do not (other than 1 and itself). If exactly one number is prime and all others are composite, that prime is the odd one out. Step-by-Step Solution:4 = 2 * 2 ⇒ composite.6 = 2 * 3 ⇒ composite.8 = 2 * 2 * 2 ⇒ composite.9 = 3 * 3 ⇒ composite.10 = 2 * 5 ⇒ composite.12 = 3 * 4 ⇒ composite.11 has no divisors other than 1 and 11 ⇒ prime. Verification / Alternative check:Among the set, only 11 cannot be expressed as a product of smaller integers. Hence it differs from all others, which are composite. Why Other Options Are Wrong:4, 6, 8, 9, 10, and 12 are all composite. They share the majority property and thus are not the odd ones out. Common Pitfalls:Mistaking 9 as prime (it is 3 * 3), or overlooking that “even vs. odd” is not the core property here because 9 is odd yet composite. The prime/composite distinction is decisive. Final Answer:11

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