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CBSE Class 7 · Mathematics

Finding Common Ground

Official NCERT chapter from Ganita Prakash Part II (book code gegp2). ExamMaster notes are original teaching at CBSE Class 7 depth.

This lesson follows the official chapter “Finding Common Ground” in Ganita Prakash Part II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 7
  • Easy level
  • 8 concepts

1The Greatest of All

The greatest common factor (HCF) is the largest whole number that divides both (or all) given numbers. “Greatest of all” here is that shared factor, not the larger of the two numbers.

The greater number is not automatically the HCF.

Figure. HCF is the greatest number that sits in both factor lists. 18 is larger, but it does not divide 12.

How it works

  1. List factors of each number (or use primes)The given pair.
  2. See the common onesThe ground.
  3. Pick the greatest of those commonsHCF.

HCF of 12 and 18

Find the HCF of 12 and 18.

  • 121,2,3,4,6,12
  • 181,2,3,6,9,18
  • Common greatest6

Pro tip. 6 divides both; 9 does not divide 12; 12 does not divide 18.

2Least, but not Last!

The least common multiple (LCM) is the smallest positive number that both (or all) divide. “Least, but not last” means smallest shared multiple — it can be larger than both.

The smaller number is not the LCM.

Figure. Multiples of 4 and of 6 first meet at 12. That least shared multiple is the LCM.

How it works

  1. List multiples, or use primesThe given pair.
  2. Find a common multipleA shared landing.
  3. Take the least positive oneLCM.

3Patterns, Properties, and a Pretty Procedure!

A useful procedure: write prime factorisations, take the least powers for HCF and the greatest powers for LCM. The pretty part is that HCF × LCM = product for two numbers (when that identity is taught).

Mixing “least powers for LCM” is a swapped procedure.

Figure. Euclid: replace the pair by (smaller, remainder) until remainder 0. The last non-zero remainder is the HCF.

How it works

  1. Factor both numbersPrimes.
  2. HCF: min powers; LCM: max powersThe procedure.
  3. Check HCF divides both and LCM is a multiple of bothThe pretty check.

4A definition is a test you can run

HCF is a test: a number d that divides both, and no larger one does. If you cannot name d and check both divisions, you do not yet have the greatest.

A common factor that is not greatest is still common — but not this heading.

Figure. 6 tiles both 12 and 18 with no leftover. 9 tiles 18 but not 12, so 9 is not the HCF.

How it works

  1. Propose a dA candidate.
  2. Check d divides bothCommon.
  3. Ask if a larger common existsGreatest or not.

5Name the given before the unknown

The given are the two (or more) numbers. The unknown is HCF or LCM. Do not start listing the larger number’s factors only.

Treating 18 as “already the answer” because it is bigger is a silent given.

Figure. Write the two numbers before you chase HCF or LCM. The ask chooses the test.

How it works

  1. Write both numbersThe given.
  2. Name HCF or LCMThe unknown.
  3. Then factor or listGiven first.

6One worked case is enough at this class

One pair is enough. A page of HCF(12,18) clones does not add a new procedure.

Ten twins of 6 are still one idea.

Figure. HCF × LCM equals the product of the two numbers. 6 × 36 = 12 × 18 = 216.

How it works

  1. Compute one HCF or one LCMThe case.
  2. A second pair only as a checkOptional.
  3. StopThis class.

7A miss: swapping the school name for the picture

A story of sharing laddoos is a setting. The maths is HCF or LCM of the two counts. Counting laddoos in a drawing and skipping the factor-test is the swap-miss.

A festive photo is not the HCF.

Figure. 18 is the largest number in sight. HCF is the greatest number that divides both.

How it works

  1. Keep the story as a settingShare, tiles, bells.
  2. Write the two numbersThe maths.
  3. Run HCF or LCMDo not swap.

8Check by the opposite action or the opposite test

Check HCF by division: both remainders 0, and a larger try fails. Check LCM: both divide it, and a smaller common multiple fails if you can find one.

Saying “6 looks common” is not the opposite test.

Figure. Check HCF by the opposite product: LCM should be 216 ÷ 6 = 36.

How it works

  1. State HCF or LCMThe claim.
  2. Run the divide-both check (and a larger/smaller challenge)The opposite test.
  3. Mend if it failsHonest.
HCF of 12 and 18 is
  1. 6
  2. 36
  3. 18

Greatest common factor, not the LCM and not the larger number.

Notes

  • The official chapter title is “Finding Common Ground”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • HCF: greatest common factor
  • LCM: least common multiple
  • For two numbers (when taught): HCF × LCM = product

Recap

Hold these pegs from the official chapter “Finding Common Ground”. The wording is ExamMaster’s teaching, not a textbook recap.

The Greatest of All
The greatest common factor (HCF) is the largest whole number that divides both (or all) given numbers.
Least, but not Last!
The least common multiple (LCM) is the smallest positive number that both (or all) divide.
Patterns, Properties, and a Pretty Procedure!
A useful procedure: write prime factorisations, take the least powers for HCF and the greatest powers for LCM.
A definition is a test you can run
HCF is a test: a number d that divides both, and no larger one does.
Name the given before the unknown
The given are the two (or more) numbers. The unknown is HCF or LCM. Do not start listing the larger number’s factors only.
One worked case is enough at this class
One pair is enough. A page of HCF(12,18) clones does not add a new procedure.

Practise Finding Common Ground

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  • A 4-question practice set that ends the chapter
  • 1 quick check with worked explanations
  • Timed mocks scored with the real marking scheme
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