NCERT Solutions for Class 7 Maths Chapter 11: Finding Common Ground – Ganita Prakash Part 2

Complete NCERT Solutions for Class 7 Maths Chapter 11 “Finding Common Ground” from the Ganita Prakash Part 2 textbook, covering HCF (Highest Common Factor), LCM (Least Common Multiple), prime factorization, and real-world applications. These Class 7 Mathematics Chapter 11 solutions are also useful as quick revision notes before exams.

3.1 The Greatest of All – Finding HCF

Tiling a room: a 12 ft × 16 ft room is to be tiled using 4 ft × 4 ft square tiles with no cutting. Since 4 divides both 12 and 16 exactly, the tiles fit perfectly: (12÷4) × (16÷4) = 3 × 4 = 12 tiles.

Bagging cement (largest equal bag size): two lots of cement, 84 kg and 108 kg, are to be packed into bags of equal weight with none left over, using the fewest bags possible — so the bag weight must be the HCF of 84 and 108. 84 = 2²×3×7 and 108 = 2²×3³; the common prime factors at their lowest powers are 2²×3 = 12 kg per bag (7 bags of 84÷12=7, and 9 bags of 108÷12=9).

Jump Jackpot (longest jump landing on both treasures): the longest jump size is the HCF of each pair — (a) 14, 30 → HCF = 2; (b) 7, 11 → HCF = 1 (co-prime, only a 1-unit jump works); (c) 30, 50 → HCF = 10; (d) 28, 42 → HCF = 14.

HCF using prime factorisation (common primes at their lowest powers):

• 24 = 2³×3, 180 = 2²×3²×5 → common = 2²×3 = 12
• 240 = 2⁴×3×5, 378 = 2×3³×7 → common = 2×3 = 6
• 400 = 2⁴×5², 2500 = 2²×5⁴ → common = 2²×5² = 100

3.2 Least, but not Last! – Finding LCM

Idli-Vada game (first shared serving number): the first number that is a multiple of both is their LCM — (a) 4, 6 → 12; (b) 7, 11 → 77; (c) 14, 30 → 210; (d) 15, 55 → 165.

LCM using prime factorisation (all primes involved, at their highest powers):

• 30 = 2×3×5, 72 = 2³×3² → LCM = 2³×3²×5 = 360
• 36 = 2²×3², 54 = 2×3³ → LCM = 2²×3³ = 108
• 105 = 3×5×7, 195 = 3×5×13, 65 = 5×13 → LCM = 3×5×7×13 = 1365
• 222 = 2×3×37, 370 = 2×5×37 → LCM = 2×3×5×37 = 1110

3.3 Patterns, Properties, and a Pretty Procedure!

General patterns: HCF of two consecutive numbers is always 1 (they are co-prime); HCF of two consecutive even numbers is always 2; LCM of two co-prime numbers always equals their product; LCM of numbers that are all multiples of 3 is itself always a multiple of 3.

HCF × LCM = product (for two numbers): for 84 and 180 — 84 = 2²×3×7, 180 = 2²×3²×5, so HCF = 2²×3 = 12 and LCM = 2²×3²×5×7 = 1260. Check: 12 × 1260 = 15,120, and 84 × 180 = 15,120 — they match, confirming the relationship.

Applications – Figure It Out (Pages 63–64)

Two numbers with HCF = 1 and LCM = 66: since the numbers are co-prime, their product equals the LCM. 6 × 11 = 66 and HCF(6,11) = 1, so a valid pair is 6 and 11.

Cowherd folklore problem: a herd of cows can be grouped exactly into rows of 3, 5, or 7, and the herd size is less than 200. The herd size must be a common multiple of 3, 5, and 7 — LCM(3,5,7) = 105. The only such multiple below 200 is 105 cows.

Cube packing: a box measuring 12 cm × 18 cm × 36 cm is to be filled exactly with equal cubes, no gaps. The cube edge must divide all three dimensions, so it must be a common factor of 12, 18, and 36 — HCF(12,18,36) = 6. So a 6 cm cube fits perfectly — (12÷6)×(18÷6)×(36÷6) = 2×3×6 = 36 cubes fill the box exactly; smaller common-factor sizes like 3 cm or 2 cm also fit exactly (with more, smaller cubes).

Largest number dividing both 306 and 36: 306 = 2×3²×17, 36 = 2²×3² → common = 2×3² = 18.

Smallest number divisible by 3, 4, 5, 7, leaving remainder 10 when divided by 11: LCM(3,4,5,7) = 420, so the number is 420k for some whole number k. 420 divided by 11 leaves remainder 2 (420 = 11×38 + 2), so 420k leaves remainder (2k mod 11); we need 2k ≡ 10 (mod 11), i.e. k ≡ 5 (mod 11). The smallest such k is 5, giving 420 × 5 = 2100. Check: 2100 ÷ 11 = 190 remainder 10 — correct.

Smallest multiple of 1, 2, 3, 4, 5, 6, 8, 9, 10 (note: 7 is not in this list): taking the highest power of each prime among these numbers (2³ from 8, 3² from 9, 5¹ from 5 or 10) gives LCM = 2³×3²×5 = 8×9×5 = 360.

Practice more: Extra Questions for Class 7 Maths Chapter 11

Quick revision: Revision Notes for Class 7 Maths Chapter 11

See the full book: NCERT Books for Class 7 Maths Part 2

Written by Satish

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