CAT Integer Solutions Bible Top 10 Must Solve Integer Solutions Questions covering 80% of Syllabus – Page 13
All 64 Integer Solutions questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 13 of 14.
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The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is
Answer: D. 5
Solution — first steps
Step 1. A nondegenerate quadrilateral exists iff its longest side is shorter than the sum of the other three.
Step 2. For sides 1, 2, 4, d this requires d<7 and 4<1+2+d.
Step 2. For sides 1, 2, 4, d this requires d<7 and 4<1+2+d.
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A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11 each, no one is left out. The maximum number of teams of 12 each that can be formed out of the students in the school is
Answer: 150
Solution — first steps
Step 1. Student count S≡4 modulo lcm(9, 10, 12, 25)=900, so S=900k+4.
Step 2. The bound S<5000 permits k = 0..5.
Step 2. The bound S<5000 permits k = 0..5.
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A donation box can receive only cheques of ₹100, ₹250, and ₹500. On one good day, the donation box was found to contain exactly 100 cheques amounting to a total sum of ₹15250. Then, the maximum possible number of cheques of ₹500 that the donation box may have contained, is
Answer: 12
Solution — first steps
Step 1. Let counts of 100, 250, 500 cheques be a, b, c, summing 100.
Step 2. Subtract 100 times the count from total value: 150b+400c=5250, or 3b+8c105.
Step 2. Subtract 100 times the count from total value: 150b+400c=5250, or 3b+8c105.
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Refer to the passage above ↑
The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is
Answer: 548
Solution — first steps
Step 1. Terms are 38+17k for integer k≥0.
Step 2. The first three-digit term occurs at k = 4: 106.
Step 2. The first three-digit term occurs at k = 4: 106.
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Suppose \(k\) is any integer such that the equation \(2x^{2} + kx + 5 = 0\) has no real roots and the equation \(x^{2} + (k - 5)x + 1 = 0\) has two distinct real roots for \(x\). Then, the number of possible values of \(k\) is
Answer: A. 9
Solution — first steps
Step 1. No real roots in first quadratic requires k²−400.
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In an examination, the average marks of students in sections A and B are 32 and 60, respectively. The number of students in section A is 10 less than that in section B. If the average marks of all the students across both the sections combined is an integer, then the difference between the maximum and minimum possible number of students in section A is
Answer: 63
Solution — first steps
Step 1. Let section A count a>0, section B count a+10.
Step 2. Combined average=(32a+60(a+10))/(2a+10)=46+70/(a+5).
Step 2. Combined average=(32a+60(a+10))/(2a+10)=46+70/(a+5).
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Consider six distinct natural numbers such that the average of the two smallest numbers is 14, and the average of the two largest numbers is 28. Then, the maximum possible value of the average of these six numbers is
Answer: D. 22.5
Solution — first steps
Step 1. Order numbers a<b<c<d<e<f. Given a+b = 28 and e+f(56).
Step 2. As e<f and integers, e≤27.
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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.