NCERT Class 7 Ganita Prakash 2 Chapter 03 Finding Common Ground PDF Download

Chapter 3: Finding Common Ground

3.1 The Greatest of All

Sameeksha is building her new house. The main room of the house is 12 ft by 16 ft. She feels that the room would look nice if the floor is covered with square tiles of the same size. She also wants to use as few tiles as possible, and for the length of the tile to be a whole number of feet. What size tiles should she buy?

Let us explore how to find the largest square tile that can be used. The breadth of the room is 12 ft and the length is 16 ft.

[Figure: Rectangle showing dimensions 16 ft by 12 ft, See in your textbook]

For the tiles to fit the breadth of the room exactly, the side of the tile should be a factor of 12. Similarly, for the tiles to fit the length of the room exactly, the side should be a factor of 16. So the side of the tile should be a factor of both 12 and 16. What are the common factors of 12 and 16?

The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 16 are 1, 2, 4, 8, 16.

The common factors are 1, 2, and 4.

So, the square tiles can have sides 1 ft, 2 ft, and 4 ft. Among these, she should use the largest sized square tile. Can you explain why?

Therefore, she needs tiles of size 4 ft.

Teacher's Note

When finding the HCF, always list all factors of each number carefully, then circle or underline which ones appear in both lists. The largest number that appears in both is your answer. In this problem, 4 is the HCF of 12 and 16, which is why Sameeksha needs 4 ft tiles - they minimize the number of tiles needed.

How many tiles of this size should she purchase?

What if Sameeksha did not insist on the length of the tile to be a whole number of feet and the length could be a fractional number of feet? Would the answer change?

The Highest Common Factor (HCF) of two or more numbers, is the highest (or greatest) of their common factors. It is also known as the Greatest Common Divisor (GCD).

In the previous problem, 4 is the HCF of 12 and 16.

We can draw rough diagrams like the one shown on the previous page to visualise the given scenario. It may help in understanding and solving.

Lekhana purchases rice from two farms and sells it in the market. She bought 84 kg of rice from one farm and 108 kg from the other farm. She wants the rice to be packed in bags, so each bag has rice from only one farm and all bags have the same weight that is a whole number of kg. If she wants to use as few bags as possible, what should the weight of each bag be?

To divide 84 kg of rice into bags of equal weight, we need the factors of 84. Similarly, for 108, we need the factors of 108.

Factors of 84 - 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

Factors of 108 - 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.

Since, Lekhana wants to use bags of the same weight for both farms, the weight of the bag should be a common factor. The common factors of 84 and 108 are

1, 2, 3, 4, 6, and 12.

She can use any of these weights to pack rice from both farms in bags of equal weight. But, she wants to minimise the number of bags.

Which weight should she choose to minimise the number of bags?

Try This

Do you remember the 'Jump Jackpot' game from Grade 6 (see the chapter 'Prime Time')? Grumpy places a treasure on a number and Jumpy chooses a jump size and tries to collect the treasure. In each case below, the two numbers upon which treasures are kept are given. Find the longest jump size (starting from 0) using which Jumpy can land on both the numbers having the treasure.

1. (a) 14 and 30
2. (b) 7 and 11
3. (c) 30 and 50
4. (d) 28 and 42

Is the longest jump size for the numbers the same as their HCF? Explain why it is so.

So far, we have been listing all the factors to find the HCF. This can become cumbersome for numbers with many factors, as you would have observed for the numbers 30 and 50, and 28 and 42. Sometimes, we may also miss some factors which can lead to errors.

Can this process be simplified? Can it be made more reliable?

It turns out that using prime factorisation can simplify the process. We will start by revisiting primes and prime factorisation.

Primes

Recall that a prime is a number greater than 1 that has only 1 and the number itself as its factors. Last year, we tried to find patterns amongst primes between 1 to 100. We also came across the Sieve of Eratosthenes - a method to list all primes.

Prime Factorisation

Any number can be written as a product of primes - keep rewriting composite factors until only primes are left.

Recall that we call this the prime factorisation of a given number. For example, we can find the prime factorisation of 90 as follows:

\[ 90 = 3 \times 3 \times 2 \times 5. \]

The number 90 could also have been factorised as \( 3 \times 30 \) or \( 2 \times 45 \) or in a few other different ways. Will these all lead to the same prime factors?

Remarkably, the resulting prime factors are always the same with perhaps only a change in their order. For example, if we consider factorising \( 3 \times 30 \) further, we get

\[ 90 = 3 \times 30 = 3 \times 2 \times 15 = 3 \times 2 \times 3 \times 5, \]

and we have arrived at the same prime factors, just in a different order.

Note that the prime factorisation of a prime number is the prime number itself.

Procedure for Prime Factorisation

Can you see what is happening below?

[Figure: Prime factorisation tree showing 105 divided by 3 to get 35, then by 5 to get 7; and 30 divided by 2 to get 15, then by 3 to get 5, See in your textbook]

Teacher's Note

Prime factorisation works by repeatedly dividing by the smallest prime that divides the number. Always start with 2, then try 3, 5, 7, and so on. For 30, you divide by 2 to get 15, then by 3 to get 5, then 5 is prime so you stop. The prime factors are 2, 3, and 5. Check your work by multiplying them back: \( 2 \times 3 \times 5 = 30 \). This method is more reliable than listing all factors by hand.

Key Points

• The Highest Common Factor (HCF) is the largest number that divides two or more numbers exactly. It helps solve real problems like finding the largest tile size or heaviest bag weight.
• To find the HCF, list all factors of each number, then pick the largest factor that appears in every list.
• Prime factorisation is writing a number as a product of primes. No matter how you start, you always get the same prime factors in the end.
• A prime is a number greater than 1 with only 1 and itself as factors. Examples are 2, 3, 5, 7, 11, 13.
• Prime factorisation is more reliable than listing all factors by hand, especially for large numbers with many factors.

This is a preview of the first 3 pages. To get the complete chapter, click below.