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

Understand the Question The given factorisations and break the numbers into composite factors, not prime factors. Just because the listed factors ( and ) do not appear identical does not mean the numbers have no common factors; composite numbers can be broken down further into common prime factors. To find the actual common factors, we find the prime factorisation of both numbers. Step 1 · Find Prime Factorisations of 72 and 144 Break down the composite factors into prime factors. For For Step 2 · Calculate the HCF Take the smallest power of each common prime factor ( and ) Smallest power of Smallest power of Since , the two numbers share multiple common factors (such as 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72). Answer > No, we cannot say that. The numbers and are composite numbers that share common prime factors ( and ), and . Common Mistakes Assuming Composite Factors are Prime: Concluding that two numbers are coprime just because the specific composite factors shown in a factorisation do not match. Not Factoring Completely: Stopping factorisation at composite numbers instead of breaking them down into prime factors.

Finding Common Ground | FIO

Question 5

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

Solution

Understand the Question

The given factorisations $72 = 6 \times 12$ and $144 = 8 \times 18$ break the numbers into composite factors, not prime factors.
Just because the listed factors ( $6, 12$ and $8, 18$ ) do not appear identical does not mean the numbers have no common factors; composite numbers can be broken down further into common prime factors.
To find the actual common factors, we find the prime factorisation of both numbers.

Step 1 · Find Prime Factorisations of 72 and 144

Break down the composite factors into prime factors. Diagram 1

For $72 = 6 \times 12$ $6 = 2 \times 3$ $12 = 2 \times 2 \times 3$

\begin{aligned} 72 &= 6 \times 12 \\ &= (2 \times 3) \times (2 \times 2 \times 3) \\ &= 2 \times 2 \times 2 \times 3 \times 3 \\ &= 2^3 \times 3^2 \end{aligned}

For $144 = 8 \times 18$ $8 = 2 \times 2 \times 2$ $18 = 2 \times 3 \times 3$

\begin{aligned} 144 &= 8 \times 18 \\ &= (2 \times 2 \times 2) \times (2 \times 3 \times 3) \\ &= 2 \times 2 \times 2 \times 2 \times 3 \times 3 \\ &= 2^4 \times 3^2 \end{aligned}

Step 2 · Calculate the HCF

Take the smallest power of each common prime factor ( $2$ and $3$ )

Smallest power of $2 = 2^3$
Smallest power of $3 = 3^2$

\begin{aligned} \text{HCF}(72, 144) &= 2^3 \times 3^2 \\ &= (2 \times 2 \times 2) \times (3 \times 3) \\ &= 8 \times 9 \\ &= 72 \end{aligned}

Since $\text{HCF}(72, 144) = 72 > 1$ , the two numbers share multiple common factors (such as 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72).

Answer

No, we cannot say that. The numbers $6, 12, 8,$ and $18$ are composite numbers that share common prime factors ( $2$ and $3$ ), and $\text{HCF}(72, 144) = 72$ .

Common Mistakes

Assuming Composite Factors are Prime: Concluding that two numbers are coprime just because the specific composite factors shown in a factorisation do not match.
Not Factoring Completely: Stopping factorisation at composite numbers instead of breaking them down into prime factors.

More questions in FIO

List all the factors of the following numbers:

(a) 90

(b) 105

(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

(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

(d) 400, 2500

(e) 300, 800

Find the LCM of the following numbers:

(a) 30, 72

(b) 36, 54

(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

(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

(d) Two co-prime numbers

Q10

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Q11

(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 $5 \times 7 \times 7 \times 11 \times 2$ ?

Q12

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

Q13

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

Q14

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.)

Q15

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

(a) $9\text{ cm}$

(b) $6\text{ cm}$

(d) $3\text{ cm}$

(e) $2\text{ cm}$

Q16

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

(a) 36

(b) 612

(d) 3

(e) 2

(f) 360

Q17

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

Q18

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

(d) 3

(e) 36

(f) None of these

Q19

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

(d) Less than $m \times n$

(e) Greater than $m \times n$

Q20

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?

Q21

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?

Q22

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