CAT Integer Solutions Bible Top 10 Must Solve Integer Solutions Questions covering 80% of Syllabus – Page 9
All 64 Integer Solutions questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 9 of 14.
Let \( n \) and \( m \) be two positive integers such that there are exactly 41 integers greater than \( 8^m \) and less than \( 8^n \), which can be expressed as powers of 2. Then, the smallest possible value of \( n + m \) is:
Answer: C. 16
Solution — first steps
Answer: C. 16 Detailed Solution: Given that there are 41 integers between \( 8^m \) and \( 8^n \), which are powers of 2.
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The population of a town in 2020 was 100,000. The population decreased by \( y\% \) from 2020 to 2021, and increased by \( x\% \) from 2021 to 2022.
If the population in 2022 was greater than in 2020 and \( x - y = 10 \), what was the lowest possible population in 2021?
If the population in 2022 was greater than in 2020 and \( x - y = 10 \), what was the lowest possible population in 2021?
Answer: B. 73,000
Solution — first steps
The lowest population in 2021 is 73,000. Using the formula for successive percentage change, the population reduces by 27%, giving \( 73,000 \) as the minimum.
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Two collectors A and B collect coins weekly in the ratio 3:4. If A collects a multiple of 7 in 5 weeks and B collects a multiple of 24 in 3 weeks, what is the minimum number of coins collected by A in one week?
Answer: B. 42
Solution — first steps
The minimum number of coins collected by A in one week is 42. The least common multiple of the constraints gives \( x = 14 \), making \( 3x = 42 \).
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Let \( a_n = 46 + 8n \) and \( b_n = 98 + 4n \) be two sequences. Find the sum of all terms common to both sequences.
Answer: C. 14,900
Solution — first steps
The sum of all common terms is 14,900. The common terms form an arithmetic sequence with 50 terms, summing to \( 14,900 \).
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The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is
Answer: 468
Solution — first steps
Step 1. A number has15 positive divisors only with prime-exponent pattern14 or(4,2), since15=15 or5×3.
Step 2. Thus the forms are p^14 or p^4q² with different primes.
Step 2. Thus the forms are p^14 or p^4q² with different primes.
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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.