Hinglish

The Greatest of All

Class 7 Maths · Chapter 11: Finding Common Ground · NCERT Ganita Prakash section P2 3.1

Learn

Meher's room is 12 ft by 16 ft. She wants equal square tiles, a whole number of feet, as few as possible. The side must divide 12 (factors 1, 2, 3, 4, 6, 12) and 16 (1, 2, 4, 8, 16). The common factors are 1, 2, 4. Fewest tiles means the biggest: 4 ft, so 3 × 4 = 12 tiles.

HCF(12, 16) = 4

The greatest common factor is the Highest Common Factor (HCF), also called the Greatest Common Divisor (GCD). Packing 84 kg and 108 kg of rice in equal bags, as few as possible: common factors 1, 2, 3, 4, 6, 12, so 12 kg bags. And the longest equal jump from 0 that lands on both 14 and 30 is HCF = 2.

HCF(84, 108) = 12

Listing factors is slow and error-prone. Prime factorisation helps: keep dividing by primes. 105 → 3 × 35 → 3 × 5 × 7. Every number has one prime factorisation (only the order can change): 90 = 3 × 30 = 3 × 2 × 15 = 2 × 3 × 3 × 5. Every factor is a sub-part of it: 840 = 2 × 2 × 2 × 3 × 5 × 7 has factor 2 × 2 × 7 = 28, since 840 = 28 × 30.

90 = 2 × 3 × 3 × 5

HCF from primes: take the primes common to both, each the fewest times it appears. 30 = 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3 share one 2 and one 3: HCF = 6. 225 = 3 × 3 × 5 × 5 and 750 = 2 × 3 × 5 × 5 × 5: HCF = 3 × 5 × 5 = 75. If no prime is shared (96 and 275), the HCF is 1. (A claim like "bigger numbers have longer factorisations" is a conjecture: 121 = 11 × 11 is a counterexample.)

HCF(30, 72) = 2 × 3 = 6

Explore: common factors.

  1. Tap every common factor of 12 and 16.

Remember

  • HCF = the greatest common factor; it answers "largest equal piece" problems.
  • From prime factorisations: common primes, each the fewest times it appears.

Quick check

What is the HCF of 45 and 75?

  1. 15
  2. 5
  3. 3
  4. 225
Answer

15. 45 = 3 × 3 × 5, 75 = 3 × 5 × 5: common 3 × 5 = 15.

Think about it

72 = 6 × 12 and 144 = 8 × 18. Does that mean their only common factor is 1?

One answer

No. Those are not prime factorisations. 72 = 2³ × 3² and 144 = 2⁴ × 3², so the HCF is 2³ × 3² = 72 (72 divides 144).

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.