Finding Common Ground

Mystery Colours!

You might have noticed and wondered about these different circle designs around the page numbers on each page! The picture below shows all the designs for the numbers from 1 to 100.

Try to decode the colour scheme for each number. There are several interesting patterns here. Share your observations with your classmates. Extending this scheme, colour the page numbers from 101 – 110.

List all the factors of the following numbers:

(a) 90

(b) 105

(c) 132

(d) 360 (this number has 24 factors)

(e) 840 (this number has 32 factors)

Find the common factors and the HCF of the following numbers:

(a) 50, 60

(b) 140, 275

(c) 77, 725

(d) 370, 592

(e) 81, 243

How do we directly find the HCF without listing all the factors?

Find the HCF of the following numbers:

(a) 24, 180

(b) 42, 75, 24

(c) 240, 378

(d) 400, 2500

(e) 300, 800

Consider the numbers 72 and 144. Suppose they are factorised into composite numbers as: 72 = 6 × 12 and 144 = 8 × 18. Seeing this, can one say that these two numbers have no common factor other than 1? Why not?

Find the LCM of the following numbers:

(a) 30, 72

(b) 36, 54

(c) 105, 195, 65

(d) 222, 370

Make a general statement about the HCF for the following pairs of numbers. You could consider examples before coming up with general statements. Look for possible explanations of why they hold.

(a) Two consecutive even numbers

(b) Two consecutive odd numbers

(c) Two even numbers

(d) Two consecutive numbers

(e) Two co-prime numbers

Share your observations with the class.

The LCM of 3 and 24 is 24 (it is one of the two given numbers).

(a) Find more such number pairs where the LCM is one of the two numbers.

(b) Make a general statement about such numbers. Describe such number pairs using algebra.

Make a general statement about the LCM for the following pairs of numbers. You could consider examples before coming up with these general statements. Look for possible explanations of why they hold.

(a) Two multiples of 3

(b) Two consecutive even numbers

(c) Two consecutive numbers

(d) Two co-prime numbers

In the two rows below, colours repeat as shown. When will the blue stars meet next?

(a) Is $5 \times 7 \times 11 \times 11$ a multiple of $5 \times 7 \times 7 \times 11 \times 2$?

(b) Is $5 \times 7 \times 11 \times 11$ a factor of $5 \times 7 \times 7 \times 11 \times 2$?

Find the HCF and LCM of the following (state your answers in the form of prime factorisations):

(a) $3 \times 3 \times 5 \times 7 \times 7$ and $12 \times 7 \times 11$

(b) 45 and 36

Find two numbers whose HCF is 1 and LCM is 66.

A cowherd took all his cows to graze in the fields. The cows came to a crossing with 3 gates. An equal number of cows passed through each gate. Later at another crossing with 5 gates again an equal number of cows passed through each gate. The same happened at the third crossing with 7 gates. If the cowherd had less than 200 cows, how many cows did he have? (Based on the folklore mathematics from Karnataka.)

The length, width, and height of a box are 12 cm, 18 cm, and 36 cm respectively. Which of the following sized cubes can be packed in this box without leaving gaps?

(a) 9 cm

(b) 6 cm

(c) 4 cm

(d) 3 cm

(e) 2 cm

Among the numbers below, which is the largest number that perfectly divides both 306 and 36?

(a) 36

(b) 612

(c) 18

(d) 3

(e) 2

(f) 360

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

Children are playing ‘Fire in the Mountain’. When the number 6 was called out, no one got out. When the number 9 was called out, no one got out. But when the number 10 was called out, some people got out. How many children could have been playing initially?

(a) 72

(b) 90

(c) 45

(d) 3

(e) 36

(f) None of these

Tick the correct statement(s). The LCM of two different prime numbers ($m, n$) can be:

(a) Less than both numbers

(b) In between the two numbers

(c) Greater than both numbers

(d) Less than $m \times n$

(e) Greater than $m \times n$

A dog is chasing a rabbit that has a head start of 150 feet. It jumps 9 feet every time the rabbit jumps 7 feet. In how many leaps does the dog catch up with the rabbit?

What is the smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 8, 9, 10? Do you remember the answer from Grade 6, Chapter 5?

Here is a problem posed by the ancient Indian Mathematician Mahaviracharya (850 C.E.). Add together $\frac{8}{15}$, $\frac{1}{20}$, $\frac{7}{36}$, $\frac{11}{63}$ and $\frac{1}{21}$. What do you get? How can we find this sum efficiently?

Context: Sameeksha is building her new house. The main room of the house is 12 ft by 16 ft. She wants to cover the floor with square tiles of the same size, using as few tiles as possible, with the length of the tile being a whole number of feet. She needs tiles of size 4 ft.

Q. 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?

Context: Lekhana bought 84 kg of rice from one farm and 108 kg from another. She wants to pack them in bags of equal weight (whole number of kg) using as few bags as possible. The common factors of 84 and 108 are 1, 2, 3, 4, 6, and 12.

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

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.

(a) 14 and 30

(b) 7 and 11

(c) 30 and 50

(d) 28 and 42

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

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

Can you see what is happening below?

Can you write the prime factorisation of 105 and 30 using these two figures?

Try finding the prime factorisation of 1200 using the method above. If we had used the earlier method, our calculation would have been as follows:

$1200 = 40 \times 30 = 5 \times 8 \times 5 \times 6 = \dots$

Which calculation is easier to carry out?

Context: Consider the number 840 and its prime factorisation $2 \times 2 \times 2 \times 3 \times 5 \times 7$.

Q. Is $2 \times 2 \times 7 = 28$ a factor of 840?

Context: Consider the number 840 and its prime factorisation $2 \times 2 \times 2 \times 3 \times 5 \times 7$.

Q. If yes, what should it be multiplied by to get 840?

Context: Consider the number 840 and its prime factorization $2 \times 2 \times 2 \times 3 \times 5 \times 7$.

Q. Similarly, is $2 \times 7 = 14$ a factor of 840? Why or why not?

Is $2 \times 2 \times 2$ a factor of 840? Why or why not?

Is $3 \times 3 \times 3$ a factor of 840? Why or why not?

Can we use this idea to list down all the possible factors of a number using just its prime factors?

Context: The factors of 225 are found to be 1, 3, 5, 9, 15, 25, 45, 75, 225.

Q. Check that all the factors of 225 occur in this list.

Do you remember the ‘Idli-Vada’ game from Grade 6 (see chapter ‘Prime Time’)? Two numbers are chosen and whenever players come to their multiples, ‘idli’ or ‘vada’ should be called out depending on whose multiple the number is. If the number happens to be a common multiple, then ‘idli-vada’ should be called out. In each problem below, the two numbers corresponding to ‘idli’ and ‘vada’ are given. Find the first number for which ‘idli-vada’ will be called out:

(a) 4 and 6

(b) 7 and 11

(c) 14 and 30

(d) 15 and 55

Is the answer always the LCM of the two numbers? Explain.

Context: Consider the numbers 14 and 35, with prime factorisations $14 = 2 \times 7$ and $35 = 5 \times 7$.

Q. Is $2 \times 3 \times 5 \times 7$ also a common multiple?

Find more such number pairs where the HCF is one of the two numbers. How can we describe such pairs of numbers?

Context: If $n$ is a number, then any multiple of $n$ can be written as a positive integer multiplied by $n$. For example, if we take $n$ and $5n$ (short for $5 \times n$), then $5n$ is a multiple of $n$, and $n$ is a factor of $5n$. The HCF of $n$ and $5n = n$.

Q. For number pairs satisfying this property (i.e., one of the numbers is the HCF),

(a) if $m$ is a number, what could be the other number?

(b) if $7k$ is a number, what could be the other number?

What happens to the HCF of two numbers if both numbers are doubled? Take some pairs of numbers and explore. Are you able to see why the HCF will also double?

Here are some more numbers where both numbers are multiples of the same number. Find their HCF:

(a) $18 \times 10$, $18 \times 15$

(b) $10 \times 38$, $10 \times 21$

(c) $5 \times 13$, $5 \times 20$

(d) $12 \times 16$, $12 \times 20$

In which of these cases is the HCF the same as the common multiplier, like problem (b) where the HCF is 10? Explore a few more examples of this type to understand when this happens.

Efficient Procedures for HCF and LCM

See the procedure on the right. Can you explain how it has been carried out?

How do we use this to find the HCF of 84 and 180? Explore.

[Hint: Observe that $84 = 2 \times 2 \times 3 \times 7$, and $180 = 2 \times 2 \times 3 \times 15$ similar to prime factorisation]

Why are these the LCMs?

[Hint: Will the product of the factors marked as the LCM of 300 and 150 contain the prime factorisations of both 300 and 150? Is this the smallest such number?]

You can try this method for these pairs of numbers.

(a) 90 and 150

(b) 84 and 132

Property Involving both the HCF and the LCM

Which is greater — the LCM of two numbers or their product?

You could analyse the above statement using examples. Then try to reason or prove, why the LCM is never greater than the product of the numbers.

[Hint: Is the product also a common multiple of the two numbers?]

Explore whether the LCM is a factor of the product in the following cases. If yes, identify the number that the LCM should be multiplied by to get the product. Do you see any pattern? Use these numbers:

(a) 45, 105

(b) 275, 352

(c) 222, 370

Context: Consider the pairs of numbers: (a) 45, 105; (b) 275, 352; (c) 222, 370.

Q. Do you see that, in each case, the number by which the LCM is multiplied to get the product is actually the HCF?

Why does this happen? Can you give an explanation or proof?

[Hint: Consider the prime factorisation of the given numbers. Among their prime factors, some are common to both factorisations, and the rest occur in only one of them. Between the HCF and the LCM, see how the common and non-common prime factors get distributed. In the product, observe how these two kinds of prime factors occur. Compare them.]

Explore whether this property holds when 3 numbers are considered.

Context: The largest prime found so far has 4,10,24,320 digits! It was discovered on October 12, 2024.

Q. If I start writing this number, how long could it take me?