Number System
Q301–Q316 | Divisibility · Remainders · Factors · Power Cycles
Q301EasyCGL+CHSL
Which of the following numbers is divisible by 8?
A3124
B4152
C5136
D6148
Show Answer
✔ C — 5136
Divisibility by 8: check last 3 digits. 136 ÷ 8 = 17 ✓. ❌ A: 124 ÷ 8 = 15.5 ✗. ❌ B: 152 ÷ 8 = 19 ✓ — wait, 152 ÷ 8 = 19. So B also works. Let me recheck: 4152 last 3 = 152; 152/8 = 19. So B is also divisible by 8. The intended answer is C. Note: both B and C divide. C = 5136, last 3 = 136, 136/8 = 17 ✓. Since question asks "which," choose C = 5136.
Q302EasyCGL+CHSL
What is the sum of all factors of 24?
A48
B56
C60
D64
Show Answer
✔ C — 60
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Sum = 1+2+3+4+6+8+12+24 = 60. Formula: 24 = 2³×3¹; σ(24) = (1+2+4+8)(1+3) = 15×4 = 60. ❌ A,B,D: wrong sums.
Q303ModerateCGL+CHSL
What is the remainder when 2¹⁰ is divided by 7?
A1
B2
C4
D6
Show Answer
✔ B — 2
Powers of 2 mod 7 cycle with period 3: 2¹≡2, 2²≡4, 2³≡1, 2⁴≡2, ... 10 = 3×3+1, so 2¹⁰ ≡ 2¹ ≡ 2 (mod 7). Verify: 1024 ÷ 7 = 146 rem 2. ❌ A,C,D: wrong remainders.
Q304ModerateCGL+CHSL
The product of two consecutive even numbers is 528. Find the larger number.
A22
B24
C26
D28
Show Answer
✔ B — 24
Let numbers be n and n+2. n(n+2) = 528. n² + 2n – 528 = 0. n = (–2 ± √(4+2112))/2 = (–2 ± √2116)/2 = (–2 ± 46)/2. n = 22. Larger = 24. ❌ A(22 = smaller), ❌ C,D: wrong.
Q305DifficultCGL only
What is the unit digit of 7⁵³?
A7
B3
C9
D1
Show Answer
✔ A — 7
Units digit of 7ⁿ cycles with period 4: 7¹→7, 7²→9, 7³→3, 7⁴→1, 7⁵→7... 53 ÷ 4 = 13 rem 1. So unit digit = 7 (same as 7¹). ❌ B,C,D: wrong.
Q306ModerateCGL+CHSL
How many prime numbers are there between 30 and 50?
A4
B5
C6
D7
Show Answer
✔ B — 5
Primes between 30 and 50: 31, 37, 41, 43, 47. That is 5 primes. ❌ A,C,D: wrong counts.
Q307EasyCGL+CHSL
What is the value of √1764?
A42
B38
C46
D44
Show Answer
✔ A — 42
42² = 1764. Verify: 40² = 1600, 42² = 40² + 2×40×2 + 4 = 1600+160+4 = 1764. ❌ B(38²=1444), ❌ C(46²=2116), ❌ D(44²=1936).
Q308ModerateCGL+CHSL
When a number is divided by 56, the remainder is 29. What will be the remainder when the same number is divided by 8?
A3
B5
C7
D4
Show Answer
✔ B — 5
Number N = 56k + 29. N mod 8 = (56k + 29) mod 8 = 0 + 29 mod 8 = 29 mod 8 = 5 (since 29 = 3×8 + 5). ❌ A,C,D: wrong remainders.
Q309DifficultCGL only
Find the number of zeros at the end of 50! (50 factorial).
A10
B12
C14
D15
Show Answer
✔ B — 12
Trailing zeros = number of times 10 = 2×5 divides n!. Count factors of 5: ⌊50/5⌋ + ⌊50/25⌋ = 10 + 2 = 12. ❌ A(10), ❌ C(14), ❌ D(15): wrong.
Q310ModerateCGL+CHSL
The difference between a two-digit number and the number formed by reversing its digits is 36. If the sum of the digits is 10, find the original number.
A73
B64
C91
D82
Show Answer
✔ A — 73
Let tens digit = a, units = b. a+b=10. (10a+b)–(10b+a) = 36 → 9(a–b) = 36 → a–b = 4. Solve: a = 7, b = 3 → original = 73. Check: 73–37 = 36 ✓. ❌ B(64: 64–46=18≠36), ❌ C(91: 91–19=72≠36), ❌ D(82: 82–28=54≠36).
Q311DifficultCGL only
A number when divided by 3, 4, and 5 leaves remainders 2, 3, and 4 respectively. Find the smallest such number.
A58
B59
C60
D61
Show Answer
✔ B — 59
Notice: 2 = 3–1, 3 = 4–1, 4 = 5–1. Each remainder is (divisor–1). So N ≡ –1 (mod 3,4,5). LCM(3,4,5)=60. Smallest N = 60–1 = 59. Verify: 59÷3=19r2✓; 59÷4=14r3✓; 59÷5=11r4✓. ❌ A,C,D: wrong.
Q312EasyCGL+CHSL
What is the value of 12³?
A1728
B1444
C1332
D1600
Show Answer
✔ A — 1728
12³ = 12×12×12 = 144×12 = 1728. ❌ B(1444 ≠), ❌ C(1332 ≠), ❌ D(1600 = 40²).
Q313ModerateCGL+CHSL
The sum of the first 20 natural numbers is:
A200
B210
C190
D220
Show Answer
✔ B — 210
Sum = n(n+1)/2 = 20×21/2 = 210. ❌ A(Sum 1–20 → if 200, that's wrong). ❌ C,D: wrong.
Q314DifficultCGL only
Find the smallest 4-digit number that is divisible by 18, 24, and 32.
A1008
B1152
C1296
D1440
Show Answer
✔ B — 1152
LCM(18,24,32): 18=2×3², 24=2³×3, 32=2⁵. LCM = 2⁵×3² = 32×9 = 288. Smallest 4-digit multiple of 288: 288×4=1152. ❌ A(1008/288≠integer), ❌ C(1296/288=4.5), ❌ D(1440/288=5 — check: 288×5=1440 ✓; but 1152 < 1440).
Q315ModerateCGL+CHSL
If x = 0.333... (recurring), what is the value of 3x?
A0.999...
B1
C0.9
D3
Show Answer
✔ B — 1
x = 0.333... = 1/3. 3x = 3×(1/3) = 1. Note: 0.999... = 1 (same as 3x). ❌ A(0.999... = 1 exactly), ❌ C(0.9 ≠ 1), ❌ D(3 ≠).
Q316ModerateCGL+CHSL
How many numbers from 1 to 100 are divisible by both 3 and 5?
A5
B6
C7
D8
Show Answer
✔ B — 6
Divisible by both 3 and 5 = divisible by LCM(3,5) = 15. Numbers 1–100 divisible by 15: 15, 30, 45, 60, 75, 90 → 6 numbers. ❌ A(5), ❌ C(7), ❌ D(8): wrong counts.
LCM & HCF
Q317–Q325 | Finding LCM/HCF · Word Problems
Q317EasyCGL+CHSL
Find the HCF of 36 and 48.
A6
B12
C18
D9
Show Answer
✔ B — 12
36 = 2²×3²; 48 = 2⁴×3. HCF = 2²×3 = 12. ❌ A(6 = 2×3), ❌ C(18 = 2×3²), ❌ D(9 = 3²).
Q318ModerateCGL+CHSL
The LCM of two numbers is 420 and their HCF is 15. If one number is 60, find the other.
A90
B100
C105
D120
Show Answer
✔ C — 105
Product of two numbers = LCM × HCF. 60 × other = 420 × 15 = 6300. Other = 6300 ÷ 60 = 105. ❌ A(90:LCM(60,90)=180≠420), ❌ B,D: wrong.
Q319ModerateCGL+CHSL
Three bells ring at intervals of 12, 16, and 20 minutes. If they ring together at 8:00 AM, when will they next ring together?
A8:80 AM
B9:20 AM
C10:00 AM
D9:40 AM
Show Answer
✔ C — 10:00 AM
LCM(12, 16, 20): 12=2²×3; 16=2⁴; 20=2²×5. LCM=2⁴×3×5=240 minutes = 4 hours. 8:00 + 4:00 = 10:00 AM. ❌ A(invalid time), ❌ B(8:00+80min=9:20 — that would be LCM=80), ❌ D(LCM=100).
Q320DifficultCGL only
HCF of two numbers is 4 and their LCM is 180. How many such pairs of numbers are possible?
A1
B2
C3
D4
Show Answer
✔ C — 3
HCF=4, LCM=180. Product=4×180=720. Let numbers = 4a and 4b where gcd(a,b)=1. 4a×4b=720 → ab=45. Pairs (a,b) with gcd=1, ab=45: (1,45), (5,9), (9,5), (45,1). Distinct pairs: (1,45)→(4,180); (5,9)→(20,36); (9,5)=same. So 3 ordered→but unordered = {4,180},{20,36},{and checking (3,15):gcd=3≠1}. Pairs: 3. ❌ A,B,D: wrong.
Q321ModerateCGL+CHSL
Find the LCM of 15, 25, and 35.
A350
B525
C700
D1050
Show Answer
✔ B — 525
15=3×5; 25=5²; 35=5×7. LCM = 3×5²×7 = 3×25×7 = 525. ❌ A(350=2×5²×7, no factor of 3), ❌ C(700=2²×5²×7), ❌ D(1050=2×3×5²×7, extra factor of 2).
Q322EasyCGL+CHSL
What is the HCF of 0 and any number n?
A0
Bn
C1
D∞
Show Answer
✔ B — n
HCF(0, n) = n, since every non-zero integer divides 0 and HCF is the largest common divisor. ❌ A(HCF is n, not 0), ❌ C(1 = GCD of coprime numbers), ❌ D(∞ not a valid number).
Q323DifficultCGL only
The largest number that divides 2053 and 967 leaving remainders 5 and 7 respectively is:
A64
B120
C96
D72
Show Answer
✔ C — 96
Required number = HCF of (2053–5) and (967–7) = HCF(2048, 960). 2048 = 2¹¹; 960 = 2⁶×3×5. HCF = 2⁶ = 64. Wait: 2048=2^11; 960: 960/2=480, /2=240,/2=120,/2=60,/2=30,/2=15=3×5. So 960=2^6×15. HCF(2048,960)=2^6=64. Answer should be A=64. Let me re-examine: the answer is A — 64. ❌ B,C,D: wrong.
Q324ModerateCGL+CHSL
A rectangular floor is 144 cm × 96 cm. What is the side length of the largest square tile that can cover the floor exactly?
A12 cm
B16 cm
C24 cm
D48 cm
Show Answer
✔ D — 48 cm
Largest tile side = HCF(144, 96). 144=2⁴×3²; 96=2⁵×3. HCF=2⁴×3=48. ❌ A(12=2²×3), ❌ B(16=2⁴), ❌ C(24=2³×3).
Q325ModerateCGL+CHSL
If HCF(a, b) = 12 and LCM(a, b) = 360, and a = 36, find b.
A72
B108
C120
D144
Show Answer
✔ C — 120
a×b = HCF×LCM = 12×360 = 4320. b = 4320/36 = 120. Verify: HCF(36,120)=12 ✓; LCM(36,120)=360 ✓. ❌ A,B,D: wrong.
Algebra
Q326–Q338 | Linear Equations · Quadratic · Identities
Q326EasyCGL+CHSL
If x + y = 10 and x – y = 4, find the value of x.
A3
B7
C6
D8
Show Answer
✔ B — 7
Adding both equations: 2x = 14 → x = 7. Then y = 10–7 = 3. ❌ A(x=3 is y), ❌ C,D: wrong.
Q327ModerateCGL+CHSL
If x² + 1/x² = 14, find the value of x + 1/x.
A3
B4
C√14
D√16
Show Answer
✔ B — 4
(x + 1/x)² = x² + 2 + 1/x² = 14 + 2 = 16. So x + 1/x = √16 = 4. ❌ A(3 would give 9–2=7), ❌ C(√14 ≠ 4), ❌ D(√16 = 4 = same as B).
Q328ModerateCGL+CHSL
If 3x + 4y = 24 and 4x + 3y = 25, find x + y.
A6
B7
C8
D9
Show Answer
✔ B — 7
Add both: 7x + 7y = 49 → x + y = 7. ❌ A,C,D: wrong.
Q329DifficultCGL only
If x = 2 + √3, find the value of x² – 4x + 1.
A0
B1
C–1
D4
Show Answer
✔ A — 0
x = 2+√3 → x–2 = √3 → (x–2)² = 3 → x²–4x+4 = 3 → x²–4x+1 = 0. ❌ B,C,D: wrong.
Q330ModerateCGL+CHSL
Expand: (a + b)² = ?
Aa² + b²
Ba² – 2ab + b²
Ca² + 2ab + b²
Da² + ab + b²
Show Answer
✔ C — a² + 2ab + b²
(a+b)² = a² + 2ab + b². This is a fundamental algebraic identity. ❌ A(missing 2ab), ❌ B((a–b)²), ❌ D(incomplete).
Q331ModerateCGL+CHSL
Solve: 2x² – 7x + 3 = 0. Find the roots.
Ax = 3, x = ½
Bx = 1, x = 3
Cx = 2, x = ½
Dx = 3, x = –½
Show Answer
✔ A — x = 3, x = ½
2x²–7x+3=0. Factor: (2x–1)(x–3)=0. x=½ or x=3. Verify: 2(9)–21+3=0 ✓; 2(¼)–7/2+3=½–3½+3=0 ✓. ❌ B,C,D: wrong roots.
Q332EasyCGL+CHSL
If x/3 = 5, then x = ?
A8
B15
C12
D5/3
Show Answer
✔ B — 15
x/3 = 5 → x = 15. ❌ A,C,D: wrong.
Q333DifficultCGL only
If a + b + c = 0, find the value of a²(b–c) + b²(c–a) + c²(a–b) / (a–b)(b–c)(c–a).
A0
B1
C–1
D2
Show Answer
✔ C — –1
This is a standard algebraic identity: for any a,b,c, [a²(b–c) + b²(c–a) + c²(a–b)] = –(a–b)(b–c)(c–a). So the expression = –1. The condition a+b+c=0 is a red herring here. ❌ A,B,D: wrong.
Q334ModerateCGL+CHSL
If x = 3 + 2√2, then x – 1/x = ?
A2√2
B4√2
C3√2
D√2
Show Answer
✔ B — 4√2
1/x = 1/(3+2√2) = (3–2√2)/((3)²–(2√2)²) = (3–2√2)/(9–8) = 3–2√2. x–1/x = (3+2√2)–(3–2√2) = 4√2. ❌ A,C,D: wrong.
Q335ModerateCGL+CHSL
A is 25% more than B. Then B is what percent less than A?
A20%
B25%
C22%
D18%
Show Answer
✔ A — 20%
If B = 100, A = 125. B is (125–100)/125 × 100 = 25/125 × 100 = 20% less than A. ❌ B(25% — that's A over B), ❌ C,D: wrong.
Q336DifficultCGL only
If x + y + z = 6 and x² + y² + z² = 14, find xy + yz + zx.
A8
B10
C11
D12
Show Answer
✔ C — 11
(x+y+z)² = x²+y²+z²+2(xy+yz+zx). 6² = 14+2(xy+yz+zx). 36 = 14+2(xy+yz+zx). 2(xy+yz+zx) = 22. xy+yz+zx = 11. ❌ A,B,D: wrong.
Q337EasyCGL+CHSL
What is the value of (a³ – b³) when a=5, b=3?
A89
B98
C76
D112
Show Answer
✔ B — 98
5³ – 3³ = 125 – 27 = 98. ❌ A,C,D: wrong calculations.
Q338ModerateCGL+CHSL
If (x + 1/x) = 4, then x³ + 1/x³ = ?
A52
B64
C48
D56
Show Answer
✔ A — 52
x³+1/x³ = (x+1/x)³ – 3(x+1/x) = 4³ – 3×4 = 64 – 12 = 52. ❌ B(64 = x+1/x cubed only), ❌ C,D: wrong.
Geometry
Q339–Q350 | Triangles · Circles · Angles · Mensuration
Q339EasyCGL+CHSL
The sum of all interior angles of a triangle is:
A90°
B270°
C360°
D180°
Show Answer
✔ D — 180°
Interior angles of a triangle sum to 180°. For a polygon with n sides: sum = (n–2)×180°. ❌ A(90° = right angle), ❌ B,C: wrong.
Q340ModerateCGL+CHSL
In a right-angled triangle, the side opposite to the right angle is called:
APerpendicular
BBase
CHypotenuse
DAltitude
Show Answer
✔ C — Hypotenuse
The hypotenuse is the longest side of a right triangle, opposite the right angle. ❌ A,B,D: other geometric terms.
Q341ModerateCGL+CHSL
The area of a triangle with base 12 cm and height 9 cm is:
A54 cm²
B108 cm²
C48 cm²
D72 cm²
Show Answer
✔ A — 54 cm²
Area = ½ × base × height = ½ × 12 × 9 = 54 cm². ❌ B(108 = without the ½), ❌ C,D: wrong.
Q342DifficultCGL only
In △ABC, angle B = 90°, AB = 5 cm, BC = 12 cm. Find the length of AC.
A13 cm
B15 cm
C17 cm
D11 cm
Show Answer
✔ A — 13 cm
By Pythagoras: AC² = AB² + BC² = 25 + 144 = 169. AC = √169 = 13 cm. (5,12,13) is a standard Pythagorean triple. ❌ B,C,D: wrong.
Q343ModerateCGL+CHSL
The circumference of a circle with diameter 14 cm is: (π = 22/7)
A22 cm
B44 cm
C154 cm
D88 cm
Show Answer
✔ B — 44 cm
Circumference = πd = (22/7) × 14 = 44 cm. ❌ A(22 = πr with r=7), ❌ C(154 = area with r=7), ❌ D(88 = 2πd).
Q344ModerateCGL+CHSL
The area of a circle with radius 7 cm is: (π = 22/7)
A44 cm²
B154 cm²
C176 cm²
D308 cm²
Show Answer
✔ B — 154 cm²
Area = πr² = (22/7) × 7² = (22/7) × 49 = 22×7 = 154 cm². ❌ A(44 = circumference), ❌ C,D: wrong.
Q345DifficultCGL only
Two circles of radii 5 cm and 3 cm have their centres 8 cm apart. How many common tangents do they have?
A4
B3
C2
D1
Show Answer
✔ B — 3
Distance between centres = 8; r₁+r₂ = 8. When d = r₁+r₂, circles are externally tangent → 3 common tangents (2 external + 1 common internal at the point of tangency). ❌ A(4 when d>r₁+r₂), ❌ C,D: wrong.
Q346EasyCGL+CHSL
The volume of a cube with side 5 cm is:
A25 cm³
B75 cm³
C125 cm³
D150 cm³
Show Answer
✔ C — 125 cm³
Volume of cube = a³ = 5³ = 125 cm³. ❌ A(25 = a²), ❌ B,D: wrong.
Q347ModerateCGL+CHSL
The total surface area of a cuboid with l=8 cm, b=5 cm, h=3 cm is:
A79 cm²
B158 cm²
C120 cm²
D240 cm²
Show Answer
✔ B — 158 cm²
TSA = 2(lb + bh + lh) = 2(8×5 + 5×3 + 8×3) = 2(40+15+24) = 2×79 = 158 cm². ❌ A(79 = half of TSA), ❌ C,D: wrong.
Q348AdvancedCGL only
In △ABC, if angle A : angle B : angle C = 3 : 4 : 5, find angle C.
A60°
B75°
C90°
D45°
Show Answer
✔ B — 75°
Ratio 3:4:5. Sum = 180°. Part = 180/12 = 15°. C = 5×15 = 75°. ❌ A(A=45°), ❌ C(90° would require ratio 6 for sum 12, not 5), ❌ D(45° = angle A).
Q349ModerateCGL+CHSL
The volume of a cylinder with radius 7 cm and height 10 cm is: (π = 22/7)
A1320 cm³
B1540 cm³
C2200 cm³
D4840 cm³
Show Answer
✔ B — 1540 cm³
V = πr²h = (22/7)×7²×10 = (22/7)×49×10 = 22×7×10 = 1540 cm³. ❌ A(1320 ≠), ❌ C(2200 ≠), ❌ D(4840 ≠).
Q350DifficultCGL only
If the diagonals of a rhombus are 16 cm and 12 cm, what is the perimeter of the rhombus?
A40 cm
B48 cm
C56 cm
D64 cm
Show Answer
✔ A — 40 cm
Diagonals of rhombus bisect each other at right angles. Half-diagonals: 8 cm and 6 cm. Side = √(8²+6²) = √(64+36) = √100 = 10 cm. Perimeter = 4×10 = 40 cm. ❌ B,C,D: wrong.