CAT Integer Solutions Bible Top 10 Must Solve Integer Solutions Questions covering 80% of Syllabus
Every real CAT question on this chapter from the past papers we have, newest year first. Passages and data sets stay together on one page.
Tap Show answer, or open the detailed solution (free login). Page 1 of 14.
Start here: the repeating CAT PYQs
Study the integer solutions pattern in each CAT PYQ, then practise variations. The 15 selected families contain 71 of the 75 questions tagged Integer constraints in the reviewed set.
CAT Integer Solutions Bible Top 10 PYQ Questions
| Questions | CAT Area Covered |
|---|---|
| First 5 CAT Questions | 46.67% |
| First 10 CAT Questions | 77.33% |
| All 15 CAT Questions | 94.67% |
Importance of Integer Solutions each year of CAT exam
| CAT year | Total Integer Solutions questions | First 6 examples | First 10 examples | All 15 examples | Outside these 15 families | Share of all Quant |
|---|---|---|---|---|---|---|
| 2021 | 9 | 4 | 7 | 9 | 0 | 13.64% |
| 2022 | 20 | 9 | 14 | 17 | 3 | 30.30% |
| 2023 | 21 | 15 | 19 | 20 | 1 | 31.82% |
| 2024 | 10 | 6 | 8 | 10 | 0 | 15.15% |
| 2025 | 15 | 7 | 10 | 15 | 0 | 22.73% |
| Total | 75 | 41 | 58 | 71 | 4 | 22.73% |
| Share of all 75 | 100% | 54.67% | 77.33% | 94.67% | 5.33% |
Quick review for beginners and last day revision
| Rule | Formula or what to do | Small example |
|---|---|---|
| Meaning of integer | Integers have no fractional part. State whether zero and negatives are allowed. | …, −2, −1, 0, 1, 2, … |
| Strict integer interval | For L < n < U, smallest n = floor(L) + 1; largest n = ceil(U) − 1. | 3 < n < 8 gives 4, 5, 6, 7. |
| Inclusive interval | For L ≤ n ≤ U, smallest n = ceil(L); largest n = floor(U). | 2.2 ≤ n ≤ 6.8 gives 3, 4, 5, 6. |
| Count allowed integers | Count = max(0, largest − smallest + 1). | Integers 4 through 9: 9 − 4 + 1 = 6. |
| Divisibility | If a count must be whole, its denominator must divide its numerator. | 140/(N + 5) is integral only if N + 5 divides 140. |
| Remainders | n ≡ r mod m means n = mk + r for an integer k. | n ≡ 2 mod 3 → n = 3k + 2 |
| Parity | Even = 2k; odd = 2k + 1. | Odd + odd = even; odd × odd = odd. |
| Factor pairs | A product equation over integers restricts the possibilities to factor pairs. | xy = 12 → positive pairs (1,12), (2,6), (3,4). |
| Distinct increasing integers | If x₁ < x₂ < … < xₙ, adjacent integers differ by at least 1. | Five distinct integers ≥ 10 have minimum sum 10 + 11 + 12 + 13 + 14. |
| Digits | A leading digit is 1 to 9; other digits are 0 to 9. | 1000a + 100b + 10c + d is a four-digit number. |
| Check an extreme | An upper bound is the answer only if a valid example attains it. | After finding n ≤ 12, verify n = 12 satisfies every condition. |
10 Core CAT PYQ Questions
Study these first.
CAT PYQ 1: Integer exponent thresholds
Understand: Use the range to find the whole numbers allowed, then check which of them give whole-number values.
Answer: 7
Solution — first steps
Step 2. The requested values 3+3^(n+1) are integers only when n≥−1.
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CAT PYQ 2: Integer count equations with totals and bounds
Understand: Write the correct, wrong and unattempted counts, use the two totals, then bound the largest count.
Answer: 24
Solution — first steps
Step 2. Subtract to obtain c−w = 11, hence w=c−11 and u = 86−2c.
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CAT PYQ 3: Prime exponent and divisor restrictions
Understand: Factor the equations. Natural numbers allow only a few factor choices.
Answer: 34
Solution — first steps
Step 2. The second gives z(x+1)=51.
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CAT PYQ 4: Ordered scores and counts under average bounds
Understand: Total = mean × number. To raise the toppers, keep the other scores as low as allowed.
Answer: 92
Solution — first steps
Step 2. To maximize each of five equal toppers, minimize the other 20 distinct integer scores with smallest 30.
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CAT PYQ 5: Integer roots and quadratic parameter bounds
Understand: Factor each equation. The common factor links a and b, then solve with natural numbers.
Answer: A. 8
Solution — first steps
Step 2. Their common positive factor gives b=2a.
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CAT PYQ 6: Factor pair enumeration with extra bounds
Understand: Match the whole-number part and the surd part, list the integer choices and take the largest.
Answer: C. 18
Solution — first steps
Step 2. Any integer representation a+b√n must have a=−3: otherwise squaring b√n=(−3−a)+2√5 would leave an irrational part in the integer b²n.
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CAT PYQ 7: AP terms intersections and index restrictions
Understand: Write the three terms as y − d, y, y + d, then use the factor pairs of the product condition.
Answer: C. 14
Solution — first steps
Step 2. Product condition gives y(y²−d²)=15y, so(y−d)(y+d)=15.
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CAT PYQ 8: Integer points under modulus or linear bounds
Understand: Compare the distances to find each boundary, then count the integers between them.
Answer: B. 19
Solution — first steps
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CAT PYQ 9: Divisibility of averages ratios and remainders
Understand: Find the total first, then pick the smallest n that makes the new average a whole odd number.
Answer: C. 5
Solution — first steps
Step 2. The new average is(39+n)/4 and must be a positive odd integer.
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CAT PYQ 10: Floor and logarithm intervals containing integers
Understand: Put log₂ n = t, find where the fraction is negative, then count the integers n in that range.
Answer: 47
Solution — first steps
Step 2. The rational expression is(4−t)/(3−t/2).
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Variations
Same idea, extra twist.
CAT PYQ 11: Digit construction and digit bounds
Understand: Write the digits as a, b, c, d, link them with the two sums, and use the 0 to 9 limits to find the highest number.
Answer: 4195
Solution — first steps
Step 2. Also c=d+4=a+5; hence b=9−2a.
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CAT PYQ 12: Integer feasibility of set overlaps
Understand: Add the set sizes. The extra over the total forces a minimum overlap; the smallest set gives the maximum.
Answer: A. 47
Solution — first steps
Step 2. Without triple overlap, 100 students can contribute at most 200 memberships; every triple student raises this cap by 1. Thus triple overlap≥5.
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CAT PYQ 13: Whole time periods and work schedules
Understand: Find the yearly gain as a fraction of the capital, then divide to get the whole years needed.
Answer: 20
Solution — first steps
Step 2. This is.012+.033333…+.004666…=.05.
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CAT PYQ 14: Distribution with minimum or even allocations
Understand: Give each child the minimum first, then share what is left for each item separately.
Answer: 1000
Solution — first steps
Step 2. Remaining allocations are 3 balloons, 3 pencils, 3 erasers among three distinct children.
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Answer: 55
Solution — first steps
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Answer: 6
Solution — first steps
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Answer: 3
Solution — first steps
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If S represents all feasible values of x, then a possible subset of S is
Answer: D. (3, √10] ∪ [5, √26) ∪ {6}
Solution — first steps
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Answer under review
Frequently asked questions
Where can I find CAT integer solutions questions with solutions?
Every Integer Solutions question on this page is a real CAT question from the CAT Actual Quant Papers. Tap Show answer for the answer; the detailed step-by-step solution opens after a free login. Start with the 15 CAT PYQs at the top, because they repeat the most.
Are there CAT integer solutions questions with answers?
Yes. Every question has its answer one tap away, free for everyone: pick an option and press Submit, or tap Show answer.
Is there a PDF of CAT integer solutions questions?
Yes. Tap Download PDF at the top of this page (free, after a quick OTP login) for the CAT Integer Solutions Bible Top 10 PYQ Questions, a free PDF of the 10 most repeated CAT integer solutions questions. All the CAT Integer Solutions questions are also on this page to read online, with answers and, after a free login, detailed solutions.
What are the CAT integer solutions previous year questions (PYQs)?
They are the real questions of the CAT Actual Quant Papers that are about integer solutions: 75 questions, grouped into 15 repeating CAT PYQs, with the other questions listed after them in paper order.
Which integer solutions questions should I practise first for CAT?
Start with the first 6 CAT PYQs, which cover about 50% of the Integer Solutions questions. Then do the first 10 (about 80%) and finally all 15 (about 90%).
How many Integer Solutions questions came in CAT papers?
64 Integer Solutions questions are on this page: 13 in CAT 2025, 10 in CAT 2024, 26 in CAT 2023, 13 in CAT 2022, 2 in CAT 2021. The same real question is shown once.
How difficult are CAT Integer Solutions questions?
Of the 12 Integer Solutions questions that carry a difficulty rating, 3 are Easy, 4 Medium and 5 Hard; the other 52 are not rated yet.
Are the solutions free?
The question and its answer are free for everyone. The detailed Cetking solution opens after a free login with your mobile number (OTP).
How is a set shown on this page?
Each passage or data set gets one page with all its questions together (4 sets here), so you never solve half of one. Questions that stand alone are shown five to a page.
Where can I practise more Integer Solutions?
Open all CAT papers and topics for every paper, or the Cetking Arena to practise.
Last day solving checklist
- Name the unknown. Include its unit and state whether it is a count, amount, rate or parameter.
- Form the relationships. Translate each given condition into an equation or restriction.
- Choose the method. Use the formula, substitution, elimination, a sign chart or a bound appropriate to this concept.
- Check restrictions. Check integer counts, signs, nonzero denominators, domains and all original conditions.
- Verify the answer. Substitute into the original question; for an extreme, show a feasible example reaches it.
All 18 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Integer exponent thresholds | 8 | CAT 2021 S2 Q15 | 8 | 10.67% |
| 2 | Integer count equations with totals and bounds | 8 | CAT 2022 S2 Q15 | 16 | 21.33% |
| 3 | Prime exponent and divisor restrictions | 7 | CAT 2022 S1 Q4 | 23 | 30.67% |
| 4 | Ordered scores and counts under average bounds | 6 | CAT 2021 S3 Q7 | 29 | 38.67% |
| 5 | Integer roots and quadratic parameter bounds | 6 | CAT 2022 S1 Q15 | 35 | 46.67% |
| 6 | Factor pair enumeration with extra bounds | 6 | CAT 2024 S1 Q2 | 41 | 54.67% |
| 7 | AP terms intersections and index restrictions | 5 | CAT 2021 S2 Q10 | 46 | 61.33% |
| 8 | Integer points under modulus or linear bounds | 4 | CAT 2021 S1 Q2 | 50 | 66.67% |
| 9 | Divisibility of averages ratios and remainders | 4 | CAT 2022 S1 Q7 | 54 | 72.00% |
| 10 | Floor and logarithm intervals containing integers | 4 | CAT 2022 S2 Q14 | 58 | 77.33% |
| 11 | Digit construction and digit bounds | 3 | CAT 2021 S2 Q12 | 61 | 81.33% |
| 12 | Integer feasibility of set overlaps | 3 | CAT 2022 S1 Q17 | 64 | 85.33% |
| 13 | Whole time periods and work schedules | 3 | CAT 2022 S2 Q18 | 67 | 89.33% |
| 14 | Distribution with minimum or even allocations | 2 | CAT 2021 S2 Q1 | 69 | 92.00% |
| 15 | Balancing distinct or near equal integers | 2 | CAT 2023 S2 Q13 | 71 | 94.67% |
| 16 | Integer bounds from ratios or polygon sides | 2 | CAT 2022 S1 Q8 | 73 | 97.33% |
| 17 | Power equals one and exponent parity | 1 | CAT 2022 S2 Q17 | 74 | 98.67% |
| 18 | Integer percentage constrained by net growth | 1 | CAT 2023 S3 Q10 | 75 | 100% |
Based on CAT Actual Quant Papers.