CAT Integer Solutions Bible Top 10 Must Solve Integer Solutions Questions covering 80% of Syllabus – Page 12
All 64 Integer Solutions questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 12 of 14.
Refer to the passage above ↑
Five students, including Amit, appear for an examination in which possible marks are integers between 0 and 50, both inclusive. The average marks for all the students is 38 and exactly three students got more than 32. If no two students got the same marks and Amit got the least marks among the five students, then the difference between the highest and lowest possible marks of Amit is
Answer: D. 20
Solution — first steps
Step 1. Total marks 190. To minimize Amit, maximize the other four as 50, 49, 48, 32.
Step 2. This gives minimum 11 and meets exactly three scores above 32.
Step 2. This gives minimum 11 and meets exactly three scores above 32.
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Refer to the passage above ↑
For some natural number n, assume that (15,000)! is divisible by (n!)!. The largest possible value of n is
Answer: B. 7
Solution — first steps
Step 1. For n = 7, n!5040≤15000, so(5040)! divides 15000!.
Step 2. For n = 8, n!40320. By Bertrand postulate there is a prime between 20160 and 40320, hence above 15000.
Step 2. For n = 8, n!40320. By Bertrand postulate there is a prime between 20160 and 40320, hence above 15000.
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Refer to the passage above ↑
The number of integer solutions of the equation (x^2 − 10)^(x^2 − 3x − 10) = 1 is
Answer: 4
Solution — first steps
Step 1. For integer x, a real integer power equals 1 via exponent 0, base 1, or base−1 with even exponent.
Step 2. Exponent x²−3x−10=0 gives x = 5 or−2, both nonzero-base cases.
Step 2. Exponent x²−3x−10=0 gives x = 5 or−2, both nonzero-base cases.
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Refer to the passage above ↑
The average of a non-decreasing sequence of N numbers a1, a2, ..., aN is 300. If a1 is replaced by 6a1, the new average becomes 400. Then, the number of possible values of a1 is
Answer: 14
Solution — first steps
Step 1. Original sum 300N. Replacing a = 1 by 6a1 increases sum by 5a1.
Step 2. The average rises 100, so 5a1=100N and a = 1=20N.
Step 2. The average rises 100, so 5a1=100N and a = 1=20N.
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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.