Class 7 Maths Part 2 Chapter 3: Finding Common Ground Notes

What is this chapter about?

Many everyday questions ask for something two numbers share. What is the biggest square tile that fits a room exactly? When will two events happen on the same day again? The first needs the highest common factor (HCF) and the second the lowest common multiple (LCM). In Class 6 you found them by listing; this chapter uses prime factorisation to find them quickly and reliably, and then looks for patterns and a short method.

Common factors and the HCF

Sameeksha wants to cover a 12 ft by 16 ft floor with equal square tiles, using as few as possible, with whole-foot sides. A tile side must divide both 12 and 16, so it must be a common factor: 1, 2 or 4. The biggest, 4 ft, needs the fewest tiles, just 3 × 4 = 12.

The HCF of two or more numbers is the highest of their common factors; it is also called the GCD (greatest common divisor). Packing 84 kg and 108 kg of rice into equal bags uses the fewest bags with 12 kg bags, since 12 is the HCF. In the Jump Jackpot game, the longest jump from 0 that lands on both treasures is also the HCF.

Prime factorisation

Listing factors is slow for big numbers and easy to get wrong. Instead, break each number into primes. The division method divides by one prime at a time, writing each quotient below, until 1 is left. A factor tree splits a number into any two factors and keeps splitting. Either way the primes are the same, only perhaps in a different order.

Every factor of a number is a part of its prime factorisation. Since 840 = 2 × 2 × 2 × 3 × 5 × 7, the part 2 × 2 × 7 = 28 is a factor, and the rest, 2 × 3 × 5 = 30, is what multiplies it to give 840. But 3 × 3 × 3 is not a factor, because 840 has only one 3. Taking every possible part lists all the factors.

Anshu guessed that a larger number always has a longer prime factorisation. Such a guess is a conjecture, and one counterexample disproves it: 96 = 2 × 2 × 2 × 2 × 2 × 3 has six primes, but the larger 121 = 11 × 11 has only two.

The HCF from prime factors

A common factor must be a part of every factorisation, so the HCF is the largest such shared part: take each common prime the fewest times it appears. For 30 = 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3, the HCF is 2 × 3 = 6. If no prime is shared, as with 96 and 275, the HCF is 1. Read more in HCF: highest common factor.

Common multiples and the LCM

Anshu’s torans are made of 6 cm strips and Guna’s of 8 cm strips. Equal torans must have a length that is a common multiple of 6 and 8, such as 24 or 48 cm, and the shortest is 24 cm. There is no largest common multiple, since doubling one gives another. Similarly, if a shop hands out gajak every Monday and Kabamai visits every 10 days, she next gets some after the LCM of 7 and 10, which is 70 days.

A multiple of a number contains all of that number’s primes, so a common multiple contains every prime of each number. The LCM keeps every prime the most times it appears: for 96 = 2 × 2 × 2 × 2 × 2 × 3 and 360 = 2 × 2 × 2 × 3 × 3 × 5, the LCM is 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 = 1440. See LCM: least common multiple.

Patterns and a quick method

If one number is a factor of the other, such as 6 and 18, the smaller is the HCF and the larger is the LCM. Consecutive numbers and co-prime numbers have HCF 1. To get both answers at once, divide the two numbers by common primes (or any common factor) in a ladder until nothing more is shared.

The divisors give the HCF, 2 × 2 × 3 = 12, and the divisors times the numbers left, 7 and 15, give the LCM, 1260. For two numbers, HCF × LCM = product of the numbers: 12 × 1260 = 15,120 = 84 × 180. Read relation between HCF and LCM. Answers to every question are in the NCERT solutions for Part 2 Chapter 3.