CAT Integer Solutions Bible Top 10 Must Solve Integer Solutions Questions covering 80% of Syllabus – Page 5
All 64 Integer Solutions questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 5 of 14.
Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value after 5 years is 25:36, then the minimum number of years required for the value to exceed Rs 20000 is _____
Answer: 9
Solution — first steps
16 → n=9
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If \((a+b \sqrt{3})^2=52+30 \sqrt{3}\), where \(a\) and \(b\) are natural numbers, then \(a+b\) equals
Answer: C. 8
Solution — first steps
Step 1. Expand (a+b√3)²=a²+3b²+2ab√3.
Step 2. Match rational and irrational parts: a²+3b²=52 and 2ab=30, so ab=15.
Step 2. Match rational and irrational parts: a²+3b²=52 and 2ab=30, so ab=15.
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In a group of 250 students, the percentage of girls was at least 44% and at most 60%. The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 80% of the girls opted for swimming while 70% of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are
Answer: 72 and 80
Solution — first steps
Step 1. Let G be girls and B=250−G be boys. The bounds give 110≤G≤150.
Step 2. Because every student chooses at least one activity, inclusion–exclusion fixes the overlap: both=0.2B+0.4G.
Step 2. Because every student chooses at least one activity, inclusion–exclusion fixes the overlap: both=0.2B+0.4G.
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If \( n \) is the least positive integer such that \( 168 \) is a factor of \( 1134^n \), and \( m \) is the least positive integer such that \( 1134^m \) is a factor of \( 168^n \), then \( m + n \) equals:
Answer: C. 15
Solution — first steps
Answer: C. 15 Detailed Solution: 1.
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Let n be the least positive integer such that 168 is a factor of 1134n. If m is the least positive integer such that 1134n is a factor of 168m, then m + n equals:
Answer: C. 15
Solution — first steps
Given 1134n and 168m conditions, solving for n and m, we get m+n = 15.
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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.