While playing the 'idli-vada' game with different number pairs, Anshu observed something interesting! 1. Sometimes the first common multiple was the same as the product of the two numbers. 2. At other times the first common multiple was less than the product of the two numbers. Find examples for each of the above. How is it related to the number pair being co-prime?

Understand the Question The first common multiple of two numbers is their Least Common Multiple (LCM). For any two positive integers and , the relationship between their LCM, HCF (Highest Common Factor), and product is given by: When two numbers are co-prime, their , which makes . When two numbers are not co-prime, their , making . Step 1 · First Common Multiple Equal to the Product Consider the number pair and . Multiples of : Multiples of : Smallest common multiple (LCM): Product of the two numbers: Since and share no common factor other than , they are co-prime. Step 2 · First Common Multiple Less than the Product Consider the number pair and . Multiples of : Multiples of : Smallest common multiple (LCM): Product of the two numbers: Since and share a common factor of (greater than ), they are not co-prime. Answer > 1. Co-prime numbers: The first common multiple equals the product of the numbers (e.g., for and , ). 2. Non-co-prime numbers: The first common multiple is strictly less than the product of the numbers (e.g., for and , ). Common Mistakes Confusing Prime with Co-prime: Believing that individual numbers must be prime. Non-prime (composite) numbers can also be co-prime if they share no common factors other than (e.g., for and , ). Missing the HCF Relationship: Forgetting that . The LCM only equals the product when .

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Question 12

While playing the 'idli-vada' game with different number pairs, Anshu observed something interesting!

Sometimes the first common multiple was the same as the product of the two numbers.
At other times the first common multiple was less than the product of the two numbers.

Find examples for each of the above. How is it related to the number pair being co-prime?

Solution

Understand the Question

The first common multiple of two numbers is their Least Common Multiple (LCM).
For any two positive integers $a$ and $b$ , the relationship between their LCM, HCF (Highest Common Factor), and product is given by: $\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b$
When two numbers are co-prime, their $\text{HCF} = 1$ , which makes $\text{LCM}(a, b) = a \times b$ .
When two numbers are not co-prime, their $\text{HCF} > 1$ , making $\text{LCM}(a, b) < a \times b$ .

Step 1 · First Common Multiple Equal to the Product

Consider the number pair $3$ and $5$ .

Multiples of $3$ : $3, 6, 9, 12, 15, 18, \dots$

Multiples of $5$ : $5, 10, 15, 20, 25, \dots$

Smallest common multiple (LCM): $\text{LCM}(3, 5) = 15$

Product of the two numbers: $3 \times 5 = 15$

$\text{LCM}(3, 5) = 3 \times 5$

Since $3$ and $5$ share no common factor other than $1$ , they are co-prime.

Step 2 · First Common Multiple Less than the Product

Consider the number pair $4$ and $6$ .

Multiples of $4$ : $4, 8, 12, 16, 20, 24, \dots$

Multiples of $6$ : $6, 12, 18, 24, 30, \dots$

Smallest common multiple (LCM): $\text{LCM}(4, 6) = 12$

Product of the two numbers: $4 \times 6 = 24$

$\text{LCM}(4, 6) < 4 \times 6 \quad (12 < 24)$

Since $4$ and $6$ share a common factor of $2$ (greater than $1$ ), they are not co-prime.

Answer

Co-prime numbers: The first common multiple equals the product of the numbers (e.g., for $3$ and $5$ , $\text{LCM} = 15 = 3 \times 5$ ).

Non-co-prime numbers: The first common multiple is strictly less than the product of the numbers (e.g., for $4$ and $6$ , $\text{LCM} = 12 < 4 \times 6$ ).

Common Mistakes

Confusing Prime with Co-prime: Believing that individual numbers must be prime. Non-prime (composite) numbers can also be co-prime if they share no common factors other than $1$ (e.g., for $8$ and $9$ , $\text{LCM}(8, 9) = 72 = 8 \times 9$ ).
Missing the HCF Relationship: Forgetting that $\text{LCM} = \dfrac{a \times b}{\text{HCF}(a, b)}$ . The LCM only equals the product when $\text{HCF} = 1$ .

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Q10

In the treasure finding game, a pair of numbers is safe if Jumpy cannot reach both using any jump size other than 1.

Q. Check if these pairs are safe:

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c. 18 and 29

d. 20 and 55

Q11

Which of the following pairs of numbers are co-prime?

(a) 18 and 35

(b) 15 and 37

(d) 17 and 69

(e) 81 and 18

Q12

While playing the 'idli-vada' game with different number pairs, Anshu observed something interesting!

Sometimes the first common multiple was the same as the product of the two numbers.
At other times the first common multiple was less than the product of the two numbers.

Find examples for each of the above. How is it related to the number pair being co-prime?

Q13

Multiply back to see that you get 36 in all four cases.

Q14

Using this diagram, can you explain why $30 = 2 \times 3 \times 5$ , no matter which way you multiply 2, 3, and 5?

Q15

Let us take $8560$ . Does it have any factors from $2$ to $10$ ( $2, 3, 4, 5, \dots, 9, 10$ )?

Q16

It is easy to check if some of these numbers are factors or not without doing long division. Can you find them?

Q17

Is 125 a multiple of 10? Will this number appear in the previous sequence? Why or why not?

Q18

Can you now answer if 8560 is divisible by 10?

Q19

Consider this statement:

Numbers that are divisible by 10 are those that end with '0'. Do you agree?

Q20

Explore by listing down the multiples: $5, 10, 15, 20, 25, \dots$ What do you observe about these numbers? Do you see a pattern in the last digit?

Q21

What is the largest number less than 399 that is divisible by 5? Is 8560 divisible by 5?

Q22

Consider this statement:

Numbers that are divisible by 5 are those that end with either a '0' or a '5'. Do you agree?

Q23

The first few multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... What do you observe? Do you see a pattern in the last digit?

Q24

Is 682 divisible by 2? Can we answer this without doing the long division?

Q25

Is $8560$ divisible by $2$ ? Why or why not?

Q26

Consider this statement:

Numbers that are divisible by 2 are those that end with '0', '2', '4', '6' or '8'. Do you agree?

What are all the multiples of 2 between 399 and 411?

Q27

Look at its multiples: 4, 8, 12, 16, 20, 24, 28, 32, ...

Are you able to observe any patterns that can be used? The multiples of 10, 5 and 2 have a pattern in their last digits which we are able to use to check for divisibility. Similarly, can we check if a number is divisible by 4 by looking at the last digit?

Q28

Can we answer the question by looking at more digits? Make a list of multiples of $4$ between $1$ and $200$ and search for a pattern.

Q29

Find numbers between 330 and 340 that are divisible by 4. Also, find numbers between 1730 and 1740, and 2030 and 2040, that are divisible by 4. What do you observe?

Q30

Is 8536 divisible by 4?

Q31

Consider these statements:

Only the last two digits matter when deciding if a given number is divisible by 4.
If the number formed by the last two digits is divisible by 4, then the original number is divisible by 4.
If the original number is divisible by 4, then the number formed by the last two digits is divisible by 4.

Do you agree? Why or why not?

Q32

Find numbers between 120 and 140 that are divisible by 8. Also find numbers between 1120 and 1140, and 3120 and 3140, that are divisible by 8. What do you observe?

Q33

Change the last two digits of 8560 so that the resulting number is a multiple of 8.

Q34

Consider these statements:

Only the last three digits matter when deciding if a given number is divisible by 8.
If the number formed by the last three digits is divisible by 8, then the original number is divisible by 8.
If the original number is divisible by 8, then the number formed by the last three digits is divisible by 8.

Do you agree? Why or why not?

Q35

There are four numbers in this box. Which number looks special to you? Why do you say so?