CAT Integer Solutions Bible Top 10 Must Solve Integer Solutions Questions covering 80% of Syllabus – Page 10
All 64 Integer Solutions questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 10 of 14.
Let \(n\) be any natural number such that \(5^{n-1} \lt 3^{n+1}\). Then, the least integer value of \(m\) that satisfies \(3^{n+1} \lt 2^{n+m}\) for each such \(n\), is
Answer: 5
Solution — first steps
Step 1. Rewrite the first inequality as (5/3)^(n−1)9. Since the base exceeds 1, allowed n are 1 through 5.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is
Answer: 340
Solution — first steps
Step 1. Let total fruit count be T; mangoes=2T/5. Write apples A and bananas B, so A+B=3T/5.
Step 2. Total sold is T/5+96+2A/5=T/2. Therefore A=3T/4−240 and B=240−3T/20.
Step 2. Total sold is T/5+96+2A/5=T/2. Therefore A=3T/4−240 and B=240−3T/20.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
The number of coins collected per week by two coin-collectors A and B are in the ratio 3 : 4. If the total number of coins collected by A in 5 weeks is a multiple of 7, and the total number of coins collected by B in 3 weeks is a multiple of 24, then the minimum possible number of coins collected by A in one week is
Answer: 42
Solution — first steps
Step 1. Let weekly collections be A=3k and B=4k.
Step 2. A's five-week total 15k is divisible by 7, so k is divisible by 7.
Step 2. A's five-week total 15k is divisible by 7, so k is divisible by 7.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
Let \(a_n=46+8 n\) and \(b_n=98+4 n\) be two sequences for natural numbers \(n \leq 100\). Then, the sum of all terms common to both the sequences is
Answer: B. 14900
Solution — first steps
Step 1. a_n=46+8n and b_n=98+4n, for 1≤n≤100.
Step 2. Every a_n is also on the b-sequence when 46+8n=98+4m, so m=2n−13.
Step 2. Every a_n is also on the b-sequence when 46+8n=98+4m, so m=2n−13.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
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.