Prime Time | IT

Question 10

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

Solution

Understand the Question

In this game, a jump size represents a common factor of both numbers.
A pair of numbers is safe if their only common factor is $1$ (i.e., they are co-prime).
If the numbers share any common factor greater than $1$ , Jumpy can reach both numbers using that jump size, making the pair not safe.

a. Check if 15 and 39 are safe.

Step 1 · Find Common Factors of 15 and 39

Diagram 1

Factors of $15$ :

\begin{aligned} 1 \times 15 &= 15 \\ 3 \times 5 &= 15 \end{aligned}

Factors of $15$ are $1, 3, 5, 15$ .

Factors of $39$ :

\begin{aligned} 1 \times 39 &= 39 \\ 3 \times 13 &= 39 \end{aligned}

Factors of $39$ are $1, 3, 13, 39$ .

Common factors of $15$ and $39$ are $1$ and $3$ .

Since they share a common factor $3 > 1$ , Jumpy can reach both using jump size $3$ .

Answer

a. The pair $15$ and $39$ is not safe.

b. Check if 4 and 15 are safe.

Step 1 · Find Common Factors of 4 and 15

Diagram 2

Factors of $4$ :

\begin{aligned} 1 \times 4 &= 4 \\ 2 \times 2 &= 4 \end{aligned}

Factors of $4$ are $1, 2, 4$ .

Factors of $15$ :

\begin{aligned} 1 \times 15 &= 15 \\ 3 \times 5 &= 15 \end{aligned}

Factors of $15$ are $1, 3, 5, 15$ .

The only common factor is $1$ .

Since there is no common factor greater than $1$ , Jumpy cannot reach both using any jump size other than $1$ .

Answer

b. The pair $4$ and $15$ is safe.

c. Check if 18 and 29 are safe.

Step 1 · Find Common Factors of 18 and 29

Diagram 3

Factors of $18$ :

\begin{aligned} 1 \times 18 &= 18 \\ 2 \times 9 &= 18 \\ 3 \times 6 &= 18 \end{aligned}

Factors of $18$ are $1, 2, 3, 6, 9, 18$ .

Factors of $29$ : $1 \times 29 = 29$ Factors of $29$ are $1, 29$ ( $29$ is a prime number).

The only common factor is $1$ .

Since there is no common factor greater than $1$ , Jumpy cannot reach both using any jump size other than $1$ .

Answer

c. The pair $18$ and $29$ is safe.

d. Check if 20 and 55 are safe.

Step 1 · Find Common Factors of 20 and 55

Diagram 4

Factors of $20$ :

\begin{aligned} 1 \times 20 &= 20 \\ 2 \times 10 &= 20 \\ 4 \times 5 &= 20 \end{aligned}

Factors of $20$ are $1, 2, 4, 5, 10, 20$ .

Factors of $55$ :

\begin{aligned} 1 \times 55 &= 55 \\ 5 \times 11 &= 55 \end{aligned}

Factors of $55$ are $1, 5, 11, 55$ .

Common factors of $20$ and $55$ are $1$ and $5$ .

Since they share a common factor $5 > 1$ , Jumpy can reach both using jump size $5$ .

Answer

d. The pair $20$ and $55$ is not safe.

Common Mistakes

Missing Factors: Forgetting to test smaller prime factors like $3$ or $5$ (e.g., missing that $39 = 3 \times 13$ ).
Confusing Prime with Co-prime: Assuming both numbers must be prime to be safe. For example, in pair $(4, 15)$ , neither number is prime, but they are co-prime ( $ext{HCF} = 1$ ), making the pair safe.
Overlooking Factor 1: Every pair shares the common factor $1$ , but the rule requires finding if any jump size other than 1 exists.

More questions in IT

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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?