CAT Integer Solutions Bible Top 10 Must Solve Integer Solutions Questions covering 80% of Syllabus – Page 3
All 64 Integer Solutions questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 3 of 14.
Let p, q and r be three natural numbers such that their sum is 900, and r is a perfect square whose values lies between 150 and 500.If p is less than 0.3q and not more than 0.7q, then the sum of the maximum and minimum possible values of p is: _____
Answer: 397
Solution — first steps
3 × 320 = 96 Pmax + P min = 301 + 96 = 397 Answer 397
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For a 4 digit number greater than 1000 , sum of the digits in thousands , hundreds and tens places is 15.And the sum of digits in hundreds , tens and units place is 16.Also the digits in tens place is 6 more than the digit in the unit place. The difference between the largest possible and smallest possible value of the number is :
Answer: D. 811
Solution — first steps
Given conditions for a 4-digit number: a + b + c = 15 —-- I b + c + d = 16 —-- II c = d + 6 —---- III Solving the equations: b = 10 - 2d a = d - 1 Now using values for d: If d = 3: a = 2, b = 4, c = 9, number = 2493 If d = 2: a = 1, b = 6, c = 8, number = 1682 Largest number = 2493, Smallest number = 1682 Difference = 2493 - 1682 = 811 Answer 811
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In a school with 1500 students , each students chooses one of the streams out of science , arts and commerce by paying a fee of rs 1100 , 1000 and 800 respectively. The total fees paid by all the students is RS. 15,50000.If the number of science students is not more than the number of arts students , then the maximum possible number of science students in school is : [TITA] _______
Answer: 700
Solution — first steps
Correct Answer: 700 Given S - science students , A - arts students , C - commerce students S + A + C = 1500 —- I 1100S + 1000A + 800C = 1550000 becomes 11S + 10A + 8C = 15500 —---- II Where S ≤ A , since we have to maximise S this S = A Multiplying I by 8 : 8S + 8A + 8C = 12000 —--- III Subtracting III from I 3S + 2A = 3500 , Since S = A 5S = 3500 , S = 700 Answer : 700
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For any natural number \( n \), let \( a_n \) be the largest integer not exceeding \( \sqrt{n} \). Then the value of \( a_1 + a_2 + \cdots + a_{50} \) is
Answer: 217
Solution — first steps
Floor of Square Roots Sum Solution We are given: For any natural number \( n \), let \( a_n = \left\lfloor \sqrt{n} \right\rfloor \) We need to find: \[ \sum_{n=1}^{50} \left\lfloor \sqrt{n} \right\rfloor \] Group the terms based on constant value of \( \left\lfloor \sqrt{n} \right\rfloor \): \( \left\lfloor \sqrt{n} \right\rfloor = k \) Range of \( n \) Count of Terms Total Sum Contributed 1 1 to 3 3 \( 1 \times 3 = 3 \) 2 4 to 8 5 \( 2 \times 5 = 10 \) 3 9 to 15 7 \( 3 \times 7 = 21 \) 4 16 to 24 9 \( 4 \times 9 = 36 \) 5 25 to 35 11 \( 5 \times 11 = 55 \) 6 36 to 48 13 \( 6 \times 13 = 78 \) 7 49 to 50 2 \( 7 \times 2 = 14 \) Now, sum all contributions: \[ 3 + 10 + 21 + 36 + 55 + 78 + 14 = \boxed{217} \] Final Answer: \[ \boxed{217} \]
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If \( m \) and \( n \) are natural numbers such that \( n > 1 \), and \( mn = 2^{25} \times 3^{40} \), then \( m - n \) equals:
Answer: 209947
Solution — first steps
Explanation Summary - mⁿ = 2²⁵ × 3⁴⁰ ✅ Explanation Summary Given: \[ m^n = 2^{25} \cdot 3^{40} \] This can be rewritten as: \[ m^n = (2^5 \cdot 3^8)^5 \] (since \( \gcd(25, 40) = 5 \), and \( n = 5 \)) Therefore: \[ m = 2^5 \cdot 3^8 = 32 \cdot 6561 = 209952 \quad \text{and} \quad n = 5 \] So, the final answer is: \[ m - n = 209952 - 5 = \boxed{209947} \]
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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.