Class 7 Maths Chapter 11 Finding Common Ground Visual Infographic | NCF 2023 Ganita Prakash
Chapter 3: Finding Common Ground
Mastering Factors, Multiples, Prime Factorisation, HCF, LCM & Fundamental Number Theorems
🔍 Highest Common Factor (HCF)
The greatest number that divides all given numbers without leaving a remainder (e.g. largest tile size for a 12 ft × 16 ft floor is 4 ft).
Take the common primes raised to their lowest (minimum) power across all factorisations.
Two numbers whose only common factor is 1 (i.e. HCF = 1, e.g. 96 and 275).
⏱️ Lowest Common Multiple (LCM)
1. What is LCM?
The smallest non-zero common multiple of given numbers (e.g. matching 6 cm & 8 cm toran strips gives LCM = 24 cm; 7-day & 10-day cycles meet at 70 days).
2. LCM by Prime Factors:
Include all prime factors that appear, raised to their highest (maximum) power.
3. Fast Simultaneous Division:
Divide numbers together by common factors until quotients are co-prime. Side column = HCF; Side × Bottom = LCM.
✨ Core Theorems & Rules
For any two positive integers a and b:
HCF(a, b) × LCM(a, b) = a × b
If a is a factor of b (e.g. n and 5n), then HCF = a and LCM = b.
If both numbers are multiplied by k, then their HCF and LCM are also multiplied by k.
🧠 Number Patterns & Historical Heritage
• Consecutive Numbers (n, n+1): Always co-prime! HCF = 1, LCM = n(n + 1).
• Consecutive Even Numbers (2n, 2n+2): Always have HCF = 2 and LCM = 2n(n + 1).
• Consecutive Odd Numbers (2n−1, 2n+1): Always co-prime! HCF = 1, LCM = product.
• Mahāvīrāchārya's Fraction Problem (850 CE): Solved fraction addition by finding the LCM of denominators (LCM of 36, 15, 20, 63, 21 = 1260).
• Prime Record Fact: The largest known prime number has 41,024,320 digits (discovered on October 12, 2024)!
⚠️ Common HCF & LCM Traps
Trap 1: The Three-Number Product Trap
HCF × LCM = Product of numbers works ONLY for two numbers. It is strictly FALSE for three or more numbers (e.g. HCF(2,4,8) × LCM(2,4,8) = 2 × 8 = 16 ≠ 64).
Trap 2: "Larger Number = Longer Factorisation" Fallacy
A larger number does NOT mean more prime factors! For example, 96 = 25 × 3 (6 prime factors), while the larger number 121 = 112 (only 2 prime factors).
