Question 1

Number series (2^n − 1 pattern): 15, 31, 63, 127, 255, (…) Determine the next Mersenne-type number in the sequence.

Accepted Answer

Introduction / Context:The sequence resembles Mersenne numbers of the form 2^n − 1. Recognizing powers of two and subtracting one is a staple pattern in number series problems. Given Data / Assumptions: Given terms: 15, 31, 63, 127, 255. Test whether each equals 2^n − 1 for increasing n. Concept / Approach:Check: 2^4 − 1 = 15, 2^5 − 1 = 31, 2^6 − 1 = 63, 2^7 − 1 = 127, 2^8 − 1 = 255. The next would be 2^9 − 1. Step-by-Step Solution: 2^4 − 1 = 16 − 1 = 15.2^5 − 1 = 32 − 1 = 31.2^6 − 1 = 64 − 1 = 63.2^7 − 1 = 128 − 1 = 127.2^8 − 1 = 256 − 1 = 255.Next: 2^9 − 1 = 512 − 1 = 511. Verification / Alternative check:Observe the near-doubling between terms, consistent with powers of two less one. Why Other Options Are Wrong: 513, 517, 523: Not equal to 2^n − 1 for n = 9; 2^9 − 1 is 511. Common Pitfalls:Assuming exact doubling; the sequence grows approximately doubling minus one, not pure geometric progression. Final Answer:511

Question 2

Number series (perfect cubes): 1, 8, 27, 64, 125, 216, (…) Identify the next cube number that continues the sequence.

Accepted Answer

Introduction / Context:This is a classic sequence of perfect cubes. Recognizing perfect powers (squares, cubes, fourth powers) is vital in numerical series questions. Given Data / Assumptions: Terms given: 1, 8, 27, 64, 125, 216. These equal 1^3, 2^3, 3^3, 4^3, 5^3, 6^3 respectively. Concept / Approach:Continue the natural-number base by one and compute the cube for the next term. Step-by-Step Solution: Identify last term's base: 216 = 6^3.Next base = 7.Compute 7^3 = 7 * 7 * 7 = 343. Verification / Alternative check:List of cubes: 1, 8, 27, 64, 125, 216, 343, 512, … The next clearly is 343. Why Other Options Are Wrong: 354, 392, 245: None are perfect cubes; they do not match 7^3. Common Pitfalls:Confusing cubes with squares or mixing arithmetic progressions into a powers sequence. Always test simple power patterns first. Final Answer:343

Question 3

Number series (two interleaved sequences): 3, 7, 6, 5, 9, 3, 12, 1, 15, (…) Find the next term by separating the odd and even positions into their own progressions.

Accepted Answer

Introduction / Context:Many series are constructed by interleaving two simpler sequences (one occupying the odd positions, the other the even). Identifying and separating them is key to determining the correct next term. Given Data / Assumptions: Total sequence: 3, 7, 6, 5, 9, 3, 12, 1, 15, (…) Odd positions: 1st, 3rd, 5th, 7th, 9th → 3, 6, 9, 12, 15. Even positions: 2nd, 4th, 6th, 8th → 7, 5, 3, 1. Concept / Approach:Treat each subsequence independently. The odd-position terms increase by +3 each step. The even-position terms decrease by −2 each step. The next term to produce is the 10th term (even position), so continue the even subsequence. Step-by-Step Solution: Odd-position subsequence: 3 → 6 → 9 → 12 → 15 (adds 3 each time).Even-position subsequence: 7 → 5 → 3 → 1 (subtracts 2 each time).Next is the 10th term (even position): 1 − 2 = −1. Verification / Alternative check:Extend both subsequences further and interleave them to confirm the original pattern is preserved. Why Other Options Are Wrong: 18: Would be the next odd-position term (after 15), not appropriate here. 13 and 3: Do not follow the even-position decrement rule from 1. Common Pitfalls:Forgetting to track whether the next required term is from the odd or even subsequence leads to choosing the wrong branch. Final Answer:-1

Question 4

Number series (identify the incorrect term): 7, 8, 18, 57, 228, 1165, 6996 Exactly one term is wrong. Select the incorrect value that breaks the pattern.

Accepted Answer

Introduction / Context:Some series hide a generative rule of the form “multiply by n and add n” with n increasing by 1 each step. The task is to detect the rule and identify which term violates it. Given Data / Assumptions: Sequence: 7, 8, 18, 57, 228, 1165, 6996. Hypothesis: a(n+1) = a(n) * k + k with k = 1, 2, 3, 4, 5, 6, … Concept / Approach:Check consecutive transitions with increasing multipliers and addends: start with ×1 + 1, then ×2 + 2, ×3 + 3, etc. Any mismatch pinpoints the incorrect term. Step-by-Step Solution: 7 → 8: 7 * 1 + 1 = 8 ✔8 → 18: 8 * 2 + 2 = 18 ✔18 → 57: 18 * 3 + 3 = 57 ✔57 → 228: Expected 57 * 4 + 4 = 228 + 4 = 232, but the term is 228 ✖If corrected to 232, then: 232 * 5 + 5 = 1165 ✔1165 * 6 + 6 = 6996 ✔ Verification / Alternative check:After fixing the fourth transition to 232, all subsequent terms align exactly with the rule ×k + k for k = 5 and k = 6, confirming 228 is the outlier. Why Other Options Are Wrong: 8, 18, 57, 1165: Each fits the rule at its step and is therefore correct. Common Pitfalls:Assuming a constant multiplier or missing the incremental +k addend. Always verify every step with the presumed rule. Final Answer:228

Question 5

Number series (find the wrong term): 1, 1, 2, 6, 24, 96, 720 Identify the single term in the sequence that does not fit the underlying pattern. Explain your reasoning clearly.

Accepted Answer

Introduction / Context:Number-series questions test the ability to spot a hidden rule. Here the sequence resembles the classic factorial pattern often used in aptitude and competitive exams. We must locate the single term that violates the rule. Given Data / Assumptions: Series: 1, 1, 2, 6, 24, 96, 720 Exactly one term is wrong. No missing terms; we are to find the misfit. Concept / Approach:Recognize factorials: n! equals the product 123*…*n. The standard factorial sequence is 1 (0!), 1 (1!), 2 (2!), 6 (3!), 24 (4!), 120 (5!), 720 (6!). Compare each given term against this benchmark. Step-by-Step Solution:Compare first five terms: 1, 1, 2, 6, 24 match 0!, 1!, 2!, 3!, 4! respectively.The sixth term should be 5! = 120, but the series shows 96.The seventh term 720 matches 6! = 720.Therefore, 96 is the single misfit; it should have been 120 to remain a proper factorial series. Verification / Alternative check:Reconstruct the corrected series: 1, 1, 2, 6, 24, 120, 720. All terms now follow n! strictly and increase as expected. Why Other Options Are Wrong: 2, 6, 24, 720: Each exactly equals 2!, 3!, 4!, and 6! respectively; these are correct. 96: Not equal to any factorial; this breaks the pattern. Common Pitfalls:Candidates sometimes assume a multiplication-by-integers rule but forget the factorial definition. Another mistake is to treat 1 as only 1!, ignoring 0! = 1, which also fits the sequence. Final Answer:96

Question 6

Number series (find the wrong term): 196, 169, 144, 121, 100, 80, 64 Exactly one term does not follow the underlying rule. Identify it and justify your choice.

Accepted Answer

Introduction / Context:This sequence looks like perfect squares listed in decreasing order. Such patterns are common in number reasoning tests. We must find the one term that is not a perfect square in the correct descending progression. Given Data / Assumptions: Series: 196, 169, 144, 121, 100, 80, 64 One term is erroneous. Concept / Approach:Recall perfect squares: 14^2 = 196, 13^2 = 169, 12^2 = 144, 11^2 = 121, 10^2 = 100, 9^2 = 81, 8^2 = 64. The intended pattern appears to be descending squares 14^2 down to 8^2. Step-by-Step Solution:Map each term to a square: 196 = 14^2 ✔, 169 = 13^2 ✔, 144 = 12^2 ✔, 121 = 11^2 ✔, 100 = 10^2 ✔.Next should be 9^2 = 81, but the series shows 80.Finally, 64 = 8^2 ✔. Therefore, 80 is the only non-square and is the wrong term. Verification / Alternative check:Replace 80 with 81 and the sequence becomes consecutive perfect squares: 196, 169, 144, 121, 100, 81, 64. Why Other Options Are Wrong: 169, 144, 121, 100: All are correct perfect squares in order. 80: Not a perfect square and disrupts the descending-square pattern. Common Pitfalls:Some candidates misremember 9^2 as 80 instead of 81 or overlook one term in a long list. Always check each term against its square root. Final Answer:80

Question 7

Number series (find the wrong term): 445, 221, 109, 46, 25, 11, 4 Spot the single incorrect term in the sequence and explain the governing rule.

Accepted Answer

Introduction / Context:Some sequences are generated by a simple linear transformation applied repeatedly. Here, the pattern involves subtracting a fixed number and dividing, a common construct in reasoning tests. We will test a consistent transform between neighboring terms. Given Data / Assumptions: Series: 445, 221, 109, 46, 25, 11, 4 Exactly one term is wrong. Concept / Approach:Try an affine step: next = (current − k) / m for constants k and m. Testing small integers quickly reveals a fit with k = 3 and m = 2. Step-by-Step Solution:From 445: (445 − 3) / 2 = 442 / 2 = 221 ✔From 221: (221 − 3) / 2 = 218 / 2 = 109 ✔From 109: (109 − 3) / 2 = 106 / 2 = 53 (but the series shows 46) ✖If we correct 46 to 53, the rule continues: from 53 → (53 − 3) / 2 = 25 ✔Then 25 → (25 − 3) / 2 = 22 / 2 = 11 ✔; and 11 → (11 − 3) / 2 = 8 / 2 = 4 ✔ Verification / Alternative check:Replacing 46 with 53 yields a perfectly consistent recurrence: next = (prev − 3) / 2 across all steps. Why Other Options Are Wrong: 221, 109, 25, 11: Each follows the transformation precisely. 46: Breaks the single consistent rule; it should be 53. Common Pitfalls:It is easy to test only addition or multiplication. Always consider mixed operations like “subtract then divide,” which commonly appear in exam sequences. Final Answer:46

Question 8

Number series (find the wrong term): 190, 166, 145, 128, 112, 100, 91 Identify the one term that violates the intended decrement pattern and justify your choice.

Accepted Answer

Introduction / Context:Many decreasing sequences are built by subtracting a series of numbers that themselves follow a simple pattern. Here the intended subtractions decrease by a constant amount each time. We must find the single term that disrupts this rhythm. Given Data / Assumptions: Series: 190, 166, 145, 128, 112, 100, 91 Exactly one term is wrong. Concept / Approach:Check consecutive differences: 190→166 (−24), 166→145 (−21), 145→128 (−17), 128→112 (−16), 112→100 (−12), 100→91 (−9). A cleaner intended pattern is to subtract numbers that drop by 3 each time: −24, −21, −18, −15, −12, −9. Step-by-Step Solution:Start with 190; subtract 24 to get 166 ✔Subtract 21 to get 145 ✔Next should subtract 18 to get 127, but the series shows 128 ✖Continue the intended pattern: 127 − 15 = 112 ✔Then 112 − 12 = 100 ✔; 100 − 9 = 91 ✔Therefore, 128 is the single inconsistent value; it should have been 127. Verification / Alternative check:The corrected series 190, 166, 145, 127, 112, 100, 91 uses differences −24, −21, −18, −15, −12, −9, a neat arithmetic-step pattern. Why Other Options Are Wrong: 166, 145, 112, 100: Each fits when differences reduce by 3 each step. 128: Deviates from the smooth −24, −21, −18, −15, −12, −9 pattern. Common Pitfalls:Overfitting irregular difference lists; instead, try to find the simplest consistent rule that fixes the sequence with a single correction. Final Answer:128

Question 9

Number series (find the wrong term): 19, 26, 33, 46, 59, 74, 91 Exactly one number does not match the series rule. Identify it and explain the rule you used.

Accepted Answer

Introduction / Context:Many series increase by repeating increments in pairs. Spotting these paired differences quickly reveals the odd term. We will analyze consecutive differences and look for a “repeat-twice then step-up” structure. Given Data / Assumptions: Series: 19, 26, 33, 46, 59, 74, 91 Only one term is to be wrong. Concept / Approach:Compute differences: +7, +7, +13, +13, +15, +17. A common intended pattern is adding an odd number twice, then moving to the next odd number twice: +7, +7, +13, +13, +15, +15, … Step-by-Step Solution:19 → 26: +7 ✔26 → 33: +7 ✔33 → 46: +13 ✔46 → 59: +13 ✔59 → 74: +15 ✔Next should repeat +15, giving 74 + 15 = 89, but the series shows 91 ✖Thus, 91 is the single inconsistent term; it should have been 89. Verification / Alternative check:Replace the last term with 89; the differences become +7, +7, +13, +13, +15, +15, which is a neat paired-odd-increments pattern. Why Other Options Are Wrong: 26, 33, 46, 59, 74: All fit the “repeat the same odd increment twice” structure. 91: Breaks the expected final +15 step; it adds +17 instead. Common Pitfalls:Assuming strictly increasing odd increments every step (+7, +9, +11, …). Here the odd increments repeat in pairs, a frequent twist in exam patterns. Final Answer:91

Question 10

Number series (find the wrong term): 1, 3, 10, 21, 64, 129, 356, 777 Find the single term that breaks the alternating multiplication pattern and justify your answer.

Accepted Answer

Introduction / Context:Alternating operations (e.g., multiply by 2 then by 3) are very common in series. This sequence strongly suggests a “multiply and add” alternation with a small constant. We will test a repeating two-step rule. Given Data / Assumptions: Series: 1, 3, 10, 21, 64, 129, 356, 777 Exactly one term is wrong. Concept / Approach:Look for an alternating pattern: 2 + 1, then 3 + 1, repeating. Verify across the sequence. Step-by-Step Solution:1 → 3: 12 + 1 = 3 ✔3 → 10: 33 + 1 = 10 ✔10 → 21: 102 + 1 = 21 ✔21 → 64: 213 + 1 = 64 ✔64 → 129: 642 + 1 = 129 ✔Next should be 1293 + 1 = 388, but the series lists 356 ✖Continuing the rule from the corrected value: 388*2 + 1 = 777, which matches the final term ✔ Verification / Alternative check:With 356 corrected to 388, every transition satisfies the alternating pattern (*2 + 1, *3 + 1), and the last term remains consistent. Why Other Options Are Wrong: 10, 21, 64, 129: Each perfectly fits the alternating rule. 356: The only value that breaks the rule; it should be 388. Common Pitfalls:Trying only constant differences or ratios; alternating mixed operations are frequently used to disguise a simple rule. Final Answer:356

Question 11

Number series (find the wrong term): 6, 12, 48, 100, 384, 768, 3072 One term violates a simple alternating multiplication pattern. Identify the wrong term and explain.

Accepted Answer

Introduction / Context:Alternating multipliers (e.g., *2, *4, *2, *4, …) generate many exam sequences. Here we will test whether such a pattern holds and isolate the single inconsistent value. Given Data / Assumptions: Series: 6, 12, 48, 100, 384, 768, 3072 Exactly one term is wrong. Concept / Approach:Try the pattern “multiply by 2, then multiply by 4,” repeating. Check each step consecutively. Step-by-Step Solution:6 → 12: *2 ✔12 → 48: *4 ✔48 → 100: should be *2 = 96, but the term is 100 ✖From 48 (corrected to 96), the pattern continues: 96 → 384 (*4) ✔384 → 768: *2 ✔; 768 → 3072: *4 ✔ Verification / Alternative check:Using 96 instead of 100 restores the exact alternating *2, *4 pattern across the entire sequence. Why Other Options Are Wrong: 12, 48, 384, 768: These align with the *2, *4 alternation. 100: The only term that does not fit; it should have been 96. Common Pitfalls:Ignoring simple alternation. Always check for repeating multipliers before trying more complex transforms. Final Answer:100

Question 12

Number series (find the wrong term): 40960, 10240, 2560, 640, 200, 40, 10 Spot the single incorrect term and describe the division pattern being followed.

Accepted Answer

Introduction / Context:Decreasing sequences are often built by repeated division by a fixed number. We will check for a consistent divisor across the list and identify the term that breaks the pattern. Given Data / Assumptions: Series: 40960, 10240, 2560, 640, 200, 40, 10 Exactly one term is wrong. Concept / Approach:Test division by 4 (a common base-2 friendly factor): 40960/4 = 10240, 10240/4 = 2560, 2560/4 = 640. Continue to see if all terms comply. Step-by-Step Solution:40960 → 10240: ÷4 ✔10240 → 2560: ÷4 ✔2560 → 640: ÷4 ✔640 → next should be 160 (÷4), but the series shows 200 ✖Continuing from the corrected 160: 160 → 40 (÷4) ✔; 40 → 10 (÷4) ✔ Verification / Alternative check:With 200 replaced by 160, the whole series becomes a perfect repeated division by 4. Why Other Options Are Wrong: 10240, 2560, 640, 40, 10: All lie on the exact ÷4 trajectory from the starting value. 200: The single value that does not equal the previous term divided by 4. Common Pitfalls:Some solvers try alternating divisors prematurely. Always check the simplest single-divisor pattern first. Final Answer:200

Question 13

Number series (find the wrong term): 3, 7, 15, 39, 63, 127, 255, 511 Determine the one term that does not belong to the underlying rule and explain why.

Accepted Answer

Introduction / Context:This sequence strongly suggests the form 2^n − 1 (Mersenne-type numbers). We will verify each term against this rule and isolate the misfit. Given Data / Assumptions: Series: 3, 7, 15, 39, 63, 127, 255, 511 Exactly one number is wrong. Concept / Approach:Compute 2^n − 1 for successive n: 2^2 − 1 = 3, 2^3 − 1 = 7, 2^4 − 1 = 15, 2^5 − 1 = 31, 2^6 − 1 = 63, 2^7 − 1 = 127, 2^8 − 1 = 255, 2^9 − 1 = 511. Step-by-Step Solution:3 ✔7 ✔15 ✔Next should be 31, but the series lists 39 ✖Then 63 ✔, 127 ✔, 255 ✔, 511 ✔ Verification / Alternative check:Replace 39 with 31 and the entire sequence becomes consecutive 2^n − 1 values for n = 2 through 9. Why Other Options Are Wrong: 7, 15, 63, 127: All equal 2^n − 1 for some n. 39: Not equal to 2^n − 1 for any integer n; thus it is the misfit. Common Pitfalls:Confusing powers of 2 minus 1 with nearby values (e.g., 31 vs. 39). Always verify by quick powers: 16, 32, 64, … Final Answer:39

Question 14

Number series (find the wrong term): 64, 71, 80, 91, 104, 119, 135, 155 One entry breaks the odd-increment pattern. Identify the wrong term and justify your reasoning.

Accepted Answer

Introduction / Context:Series formed by adding successive odd numbers are very common. Here, the increments appear to grow by 2 each time. We must detect the single place where the increment is not the expected odd step. Given Data / Assumptions: Series: 64, 71, 80, 91, 104, 119, 135, 155 Exactly one term is wrong. Concept / Approach:Compute the differences: 71−64 = 7, 80−71 = 9, 91−80 = 11, 104−91 = 13, 119−104 = 15, 135−119 = 16, 155−135 = 20. The intended pattern is adding consecutive odd numbers: +7, +9, +11, +13, +15, +17, +19. Step-by-Step Solution:Up to 119, the increments are 7, 9, 11, 13, 15 ✔Next should be +17, so 119 + 17 = 136, but the series shows 135 ✖Finally, the next increment should be +19, and from the corrected 136, 136 + 19 = 155 ✔Thus, only 135 is inconsistent; it should have been 136. Verification / Alternative check:Replace 135 with 136; the full list of differences becomes +7, +9, +11, +13, +15, +17, +19, a perfect run of consecutive odd numbers. Why Other Options Are Wrong: 71, 80, 104, 119: They fit the expected odd increments sequence. 135: Breaks the odd-number increment pattern by using +16 instead of +17. Common Pitfalls:Overlooking a single off-by-one error among many correct odd increments. Always compute all consecutive differences. Final Answer:135

Question 15

Number series – find the incorrect term: 15, 16, 34, 105, 424, 2124, 12756

Accepted Answer

Introduction / Context:This problem tests recognition of multiplicative patterns with a small additive tweak in an integer sequence. Your goal is to detect which term breaks the otherwise consistent rule governing the series. Given Data / Assumptions: Series: 15, 16, 34, 105, 424, 2124, 12756 Exactly one term is wrong. Operations may involve multiplying by a changing factor and adding a matching increment. Concept / Approach: Look for a pattern such as “multiply by n, then add n” where n increases by 1 each step. This type of linear-increment multiplier pattern is common in reasoning tests. Step-by-Step Solution: Start with 15: 151 + 1 = 16 ✓Next: 162 + 2 = 34 ✓Next: 343 + 3 = 105 ✓Next: 1054 + 4 = 424 ✓Next should be: 4245 + 5 = 2125 (not 2124) ✗Continue check: 21256 + 6 = 12756 ✓ Verification / Alternative check: All transitions obey the rule multiply-by-n then add n (n = 1, 2, 3, 4, 5, 6) except the given 2124. Replacing it with 2125 restores perfect consistency. Why Other Options Are Wrong: 16, 34, 105, 424, and 12756 each exactly satisfy the rule with their neighbors, so they are not erroneous. Common Pitfalls: Trying constant differences or ratios will mislead; the key is the steadily increasing multiplier and the matched addend. Final Answer: 2124

Question 16

Number series – find the incorrect term: 10, 26, 74, 218, 654, 1946, 5834

Accepted Answer

Introduction / Context:This sequence is governed by a single, repeated operation across all terms. Your task is to identify the one number that violates the rule. Given Data / Assumptions: Series: 10, 26, 74, 218, 654, 1946, 5834 Exactly one term is wrong. Operations likely involve consistent multiplication with a fixed adjustment. Concept / Approach: Try a linear transform of the form “next = current3 - 4”. These compact rules are common in reasoning sequences. Step-by-Step Solution: 103 - 4 = 26 ✓263 - 4 = 74 ✓743 - 4 = 218 ✓2183 - 4 = 650 (but the list shows 654) ✗If corrected to 650, then 6503 - 4 = 1946 ✓1946*3 - 4 = 5834 ✓ Verification / Alternative check: Replacing 654 with 650 makes the relation exact for every step. No other single substitution yields universal consistency. Why Other Options Are Wrong: 26, 74, 218, 1946 all satisfy the rule with neighbors; they are internally consistent. Common Pitfalls: Do not assume changing multipliers; a fixed multiplier plus a constant offset is sufficient here. Final Answer: 654

Question 17

Number series – find the incorrect term: 2880, 480, 92, 24, 8, 4, 4

Accepted Answer

Introduction / Context:Here the pattern is a clean sequence of successive divisions by descending integers. One term breaks this elegant structure. Given Data / Assumptions: Series: 2880, 480, 92, 24, 8, 4, 4 Exactly one term is wrong. Look for a factorial-like cascade or division by 6, then 5, then 4, etc. Concept / Approach: Check the idea “divide by 6, 5, 4, 3, 2, 1”. Such a descending-divisor pattern is a standard test item. Step-by-Step Solution: 2880 / 6 = 480 ✓480 / 5 = 96 (but the series shows 92) ✗96 / 4 = 24 ✓24 / 3 = 8 ✓8 / 2 = 4 ✓4 / 1 = 4 ✓ Verification / Alternative check: If we replace 92 with 96, the “divide by 6,5,4,3,2,1” rule holds perfectly for every step. Why Other Options Are Wrong: 480, 24, 8, 4 are completely consistent with the corrected chain of divisions. Common Pitfalls: Attempting to fit mixed add/subtract steps obscures the far simpler division pattern. Final Answer: 92

Question 18

Number series – find the incorrect term: 3, 7, 15, 27, 63, 127, 255

Accepted Answer

Introduction / Context:This is a classic “Mersenne-like” sequence test where terms are one less than powers of 2. One value does not fit that model. Given Data / Assumptions: Series: 3, 7, 15, 27, 63, 127, 255 Exactly one term is wrong. Check against 2^n - 1 structure. Concept / Approach: Compute 2^n - 1 for n = 2, 3, 4, 5, 6, 7, 8 to see what the sequence should be. Step-by-Step Solution: 2^2 - 1 = 3 ✓2^3 - 1 = 7 ✓2^4 - 1 = 15 ✓2^5 - 1 = 31 (but the series shows 27) ✗2^6 - 1 = 63 ✓2^7 - 1 = 127 ✓2^8 - 1 = 255 ✓ Verification / Alternative check: Replacing 27 with 31 restores the exact 2^n - 1 progression across all terms. Why Other Options Are Wrong: 7, 15, 63, 127 match perfect Mersenne form with their respective exponents and are correct. Common Pitfalls: Looking for differences obscures the simpler power-of-two relationship. Final Answer: 27

Question 19

Find the odd number out in the sequence: 835, 734, 642, 751, 853, 981, 532 (each is a three-digit number).

Accepted Answer

Introduction / Context:Odd-man-out puzzles often hide a digit-wise rule within three-digit numbers. A common pattern uses the middle digit as a function (sum or difference) of the first and last digits. We examine each number to see whether its tens digit equals the difference between its hundreds and units digits. Given Data / Assumptions: Sequence: 835, 734, 642, 751, 853, 981, 532 Hypothesis: For a valid term, tens digit = hundreds digit − units digit Concept / Approach:Check each term digit-wise. If hundreds − units equals the tens digit, it fits the rule. Otherwise it is the outlier. This is faster than seeking arithmetic progressions or sums across the whole numbers. Step-by-Step Solution:835: 8 − 5 = 3 ⇒ tens = 3 ✔734: 7 − 4 = 3 ⇒ tens = 3 ✔642: 6 − 2 = 4 ⇒ tens = 4 ✔751: 7 − 1 = 6 ⇒ tens shown is 5 ✖853: 8 − 3 = 5 ⇒ tens = 5 ✔981: 9 − 1 = 8 ⇒ tens = 8 ✔532: 5 − 2 = 3 ⇒ tens = 3 ✔ Verification / Alternative check:All except 751 satisfy the difference rule exactly; no other hidden constraint (such as sum equality) fits the set as neatly, confirming 751 as the unique mismatch. Why Other Options Are Wrong:853/981/532 each obey the pattern tens = hundreds − units and are therefore not odd. Common Pitfalls:Trying total sums or prime checks on whole numbers rather than exploring intra-digit rules; overlooking that subtraction order matters (hundreds minus units, not vice versa). Final Answer:751

Question 20

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

Accepted Answer

Introduction / Context:This classic odd-one-out set contains well-known perfect cubes. The task is to identify which term is not a cube of a positive integer. Perfect-power recognition (especially squares and cubes) is a common aptitude skill. Given Data / Assumptions: 8 = 2^3, 27 = 3^3, 64 = 4^3 125 = 5^3, 216 = 6^3, 343 = 7^3 100 is not an integer cube Concept / Approach:Verify each term against n^3 for small integers n. Since the neighbors match consecutive cubes (2 through 7 with a gap), the non-cube stands out immediately. Step-by-Step Solution:List cubes: 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125, 6^3 = 216, 7^3 = 343Compare: every term except 100 coincides with one of these cubesHence, 100 is not a perfect cube ⇒ odd term Verification / Alternative check:Try cube roots: ∛100 ≈ 4.64 (not integer), whereas ∛125 = 5 and ∛64 = 4, confirming uniqueness of 100. Why Other Options Are Wrong:27/125/343 are exact cubes and therefore fit the intended pattern. Common Pitfalls:Confusing 100 with the nearby cube 125; misreading 216 as 6^2 instead of 6^3 (it is 6^3). Final Answer:100

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