3. Finding Common Ground

7 topics1 h

Chapter 3 of NCERT Mathematics for Class 7: Finding Common Ground. Three sections — Prime factorisation, HCF and LCM from the factorisations and Patterns, properties, and a faster procedure. Seven videos, 1 h in all.

Worked answers to the 21 exercise questions in this chapter →

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Part 1

Prime factorisation

10 minBreaking a number down to primes, and why the result is uniquePull 90 apart wherever you like — 3 × 30, or 2 × 45, or 9 × 10. Every route ends at the same handful of primes.10 min · 12 partsहिंदी10 minReading every factor of a number off its prime factorisationIs 28 a factor of 840? You could divide and see. That works, and it teaches you nothing.10 min · 12 partsहिंदी

Part 2

HCF and LCM from the factorisations

10 minHCF: take the fewest occurrences of each primeA room 12 ft by 16 ft, square tiles, whole feet, as few as possible. A 3 ft tile fits the 12 and leaves a foot over on the 16.10 min · 12 partsहिंदी11 minLCM: take the most occurrences of each primeTwo paper chains, one of 6 cm links and one of 8 cm, whole links only, both finishing at the same length.11 min · 12 partsहिंदी

Part 3

Patterns, properties, and a faster procedure

11 minWhat HCF and LCM do for consecutive, even, and co-prime numbers6 and 18 have an HCF of 6 — and 6 was one of the numbers we started with. The answer landed on the pair, not between them.11 min · 12 partsहिंदी10 minGetting the HCF and the LCM out of one division ladderA layout arrives with nothing on it labelled: 84 and 180 at the top, some numbers down the left, a closing row underneath.10 min · 12 partsहिंदी10 minConjecture and generalisation: what mathematicians mean by those wordsThis one is not about factors. It is about the status a claim carries, and about saying which status you are claiming.10 min · 12 partsहिंदी

What you will be able to do

  • Say why writing out a number's factors one by one is unreliable, and name the two things that go wrong with it
  • Decide whether a stated product of primes divides a stated number, by looking at its factorisation instead of by dividing
  • Recognise a "largest equal piece" problem and say why it is a common-factor problem
  • Recognise a "when do two repeating things line up" problem and say why it is a common-multiple problem
  • Describe, without listing, which pairs have an HCF equal to one of the pair itself, and do the same for the LCM
  • Read a stacked-division layout and say what each row and each divisor records
  • Say what makes a claim a conjecture rather than an established fact

What this chapter assumes you already know

  • Factor and multiple as they were used in Class 6 (Prime Time): a factor divides exactly, leaving nothing over
  • What a prime number is, and the Sieve of Eratosthenes as a way of listing primes — both are recalled here rather than built
  • Multiplying three or four small numbers together in any order and getting the same product
  • Rearranging a product without changing its value
  • Reading a product of several primes as a single number, and back again
  • Class 6 Prime Time: a factor divides exactly
  • Class 6 Prime Time: common factors, and the Jump Jackpot game on a number line
  • Area of a rectangle as rows times columns, for the tiling problem

It builds directly on Factorise and regroup instead of multiplying head-on.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "You have to start with the smallest prime, or you will get the wrong answer."
  • "Different splittings give different factorisations."
  • "The chapter proves the factorisation is the same every time."
  • "A prime has no prime factorisation."
  • "Bigger numbers take longer to factorise."
  • "The circled diagram and the ruled layout are two different methods."
  • "Every subpart of the factorisation is a factor — so 3 × 3 × 3 is a factor of 840."
  • "You could still find a factor that is not made of the number's own primes."