Look at the table below. What do you notice? In the table, 1. Is there anything common among the shaded numbers? 2. Is there anything common among the circled numbers? 3. Which numbers are both shaded and circled? What are these numbers called?

Understand the Question We observe two visual patterns across the numbers in the table: Shaded numbers increase by each time, representing consecutive multiples of . Circled numbers increase by each time, representing consecutive multiples of . Numbers that are both shaded and circled belong to both patterns at the same time, which makes them common multiples of and (multiples of ). Part 1. Is there anything common among the shaded numbers? Step 1 · Identify Pattern in Shaded Numbers List of shaded numbers: Finding the difference between consecutive shaded numbers: Each shaded number is more than the previous number. Therefore, all shaded numbers are multiples of starting from . Answer > 1. The shaded numbers are multiples of starting from . Part 2. Is there anything common among the circled numbers? Step 1 · Identify Pattern in Circled Numbers List of circled numbers: Finding the difference between consecutive circled numbers: Each circled number is more than the previous number. Therefore, all circled numbers are multiples of starting from . Answer > 2. The circled numbers are multiples of starting from . Part 3. Which numbers are both shaded and circled? What are these numbers called? Step 1 · Find Common Numbers Comparing the two lists: Shaded numbers (multiples of ): Circled numbers (multiples of ): The numbers present in both lists are , and . Since they are multiples of both and , they are called common multiples of and . Answer > 3. The numbers both shaded and circled are , and . These are called common multiples of and . Common Mistakes Confusing Factors and Multiples: Multiples are generated by multiplying the number by integers (), whereas factors are numbers that divide evenly into it. Incomplete Verification: Assuming any even multiple of is also a multiple of (for instance, and are multiples of and even, but are not divisible by ).

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Question 5

Look at the table below. What do you notice?

In the table,

Is there anything common among the shaded numbers?
Is there anything common among the circled numbers?
Which numbers are both shaded and circled? What are these numbers called?

Solution

Understand the Question

We observe two visual patterns across the numbers in the table:
- Shaded numbers increase by $3$ each time, representing consecutive multiples of $3$ .
- Circled numbers increase by $4$ each time, representing consecutive multiples of $4$ .
Numbers that are both shaded and circled belong to both patterns at the same time, which makes them common multiples of $3$ and $4$ (multiples of $\text{LCM}(3, 4) = 12$ ).

1. Is there anything common among the shaded numbers?

Step 1 · Identify Pattern in Shaded Numbers

List of shaded numbers: $33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69$ Diagram 1

Finding the difference between consecutive shaded numbers:

\begin{aligned} 36 - 33 &= 3 \\ 39 - 36 &= 3 \\ 42 - 39 &= 3 \end{aligned}

Each shaded number is $3$ more than the previous number. Therefore, all shaded numbers are multiples of $3$ starting from $33$ .

Answer

1. The shaded numbers are multiples of $3$ starting from $33$ .

2. Is there anything common among the circled numbers?

Step 1 · Identify Pattern in Circled Numbers

List of circled numbers: $32, 36, 40, 44, 48, 52, 56, 60, 64, 68$ Diagram 2

Finding the difference between consecutive circled numbers:

\begin{aligned} 36 - 32 &= 4 \\ 40 - 36 &= 4 \\ 44 - 40 &= 4 \end{aligned}

Each circled number is $4$ more than the previous number. Therefore, all circled numbers are multiples of $4$ starting from $32$ .

Answer

2. The circled numbers are multiples of $4$ starting from $32$ .

3. Which numbers are both shaded and circled? What are these numbers called?

Step 1 · Find Common Numbers

Comparing the two lists:

Shaded numbers (multiples of $3$ ): $33, \mathbf{36}, 39, 42, 45, \mathbf{48}, 51, 54, 57, \mathbf{60}, 63, 66, 69$
Circled numbers (multiples of $4$ ): $32, \mathbf{36}, 40, 44, \mathbf{48}, 52, 56, \mathbf{60}, 64, 68$

The numbers present in both lists are $36, 48$ , and $60$ . Since they are multiples of both $3$ and $4$ , they are called common multiples of $3$ and $4$ .

Answer

3. The numbers both shaded and circled are $36, 48$ , and $60$ . These are called common multiples of $3$ and $4$ .

Common Mistakes

Confusing Factors and Multiples: Multiples are generated by multiplying the number by integers ( $3, 6, 9, \dots$ ), whereas factors are numbers that divide evenly into it.
Incomplete Verification: Assuming any even multiple of $3$ is also a multiple of $4$ (for instance, $42$ and $54$ are multiples of $3$ and even, but are not divisible by $4$ ).

More questions in IT

Find out other such numbers that are multiples of both 3 and 5. These numbers are called ____________________.

Yesterday, we played this game with two numbers. We ended up saying just 'idli' or 'idli-vada' and nobody said just 'vada'! One of the numbers was 4. Oh, what could those numbers be!?

Which of the following could be the other number: 2, 3, 5, 8, 10?

Yesterday, we played this game with two numbers. We ended up saying just 'idli' or 'idli-vada' and nobody said just 'vada'! One of the numbers was 4.

Q. Which of the following could be the other number: 2, 3, 5, 8, 10?

What jump size can reach both 15 and 30? There are multiple jump sizes possible. Try to find them all.

Look at the table below. What do you notice?

In the table,

Is there anything common among the shaded numbers?
Is there anything common among the circled numbers?
Which numbers are both shaded and circled? What are these numbers called?

How many arrangements are possible? Think and find out the different ways how—

Guna can arrange 12 figs in a rectangular manner.
Anshu can arrange 7 figs in a rectangular manner.

Observe the number of rows and columns in each of the arrangements. How are they related to 12?

How many prime numbers are there from 21 to 30? How many composite numbers are there from 21 to 30?

In the treasure finding game, treasures are kept on two numbers. Jumpy gets the treasures only if he is able to reach both the numbers with the same jump size. A jump size of 1 is not allowed.

Q. Where should Grumpy place the treasures so that Jumpy cannot reach both the treasures?

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:

a. 15 and 39

b. 4 and 15

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?