Patterns, Properties, and a Pretty Procedure!

Class 7 Maths · Chapter 11: Finding Common Ground · NCERT Ganita Prakash section P2 3.3

Learn

General statements: the HCF of 6 and 18 is 6, because 6 is a factor of 18. In general the HCF of n and 5n is n: "be my factor, I'll be your multiple!" Two consecutive numbers have HCF 1; two consecutive even numbers have HCF 2. Numbers with HCF 1 are co-prime, and their LCM is their product.

HCF(n, 5n) = n

Doubling both numbers doubles the HCF: 270 and 50 have HCF 10, while 540 and 100 have HCF 20 (each gets one more 2). Multiples of the same number: 14 × 6 and 14 × 9 have HCF 14 × 3 = 42, not just 14, because 6 and 9 share a 3.

HCF(14 × 6, 14 × 9) = 42

A pretty procedure: divide both numbers by a common prime again and again: 84, 180 → 42, 90 → 21, 45 → 7, 15 (no common prime left). HCF = 2 × 2 × 3 = 12, the divisors on the left. The LCM is the divisors times the last row: 12 × 7 × 15 = 1260. Any common factor can be used in a step, like dividing 630, 770 by 10 first.

HCF = 12 · LCM = 12 × 7 × 15 = 1260

HCF × LCM = product. For 105 = 3 × 5 × 7 and 95 = 5 × 19: LCM = 3 × 5 × 7 × 19, HCF = 5, and 105 × 95 = LCM × 5. The shared primes go once into the HCF and once into the LCM; the rest go into the LCM. So the LCM is never more than the product.

HCF × LCM = a × b

Explore: co-prime pairs.

1. Sort the pairs: is the HCF 1?

Remember

• Consecutive numbers are co-prime; the LCM of co-prime numbers is their product.
• HCF × LCM equals the product of the two numbers.

Quick check

Two numbers have HCF 6 and LCM 180. One is 36. What is the other?

1. 30
2. 6
3. 180
4. 216

Think about it

Find the smallest number divisible by 3, 4, 5 and 7 that leaves remainder 10 when divided by 11.

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.