Prime Time

Let us now play the 'idli-vada' game with different pairs of numbers:

a. 2 and 5, b. 3 and 7, c. 4 and 6.

We will say 'idli' for multiples of the smaller number, 'vada' for multiples of the larger number and 'idli-vada' for common multiples. Draw a figure similar to Fig. 5.1 if the game is played up to 60.

Co-prime art

Observe the following thread art. The first diagram has 12 pegs and the thread is tied to every fourth peg (we say that the thread-gap is 4). The second diagram has 13 pegs and the thread-gap is 3. What about the other diagrams? Observe these pictures, share and discuss your findings in class.

In some diagrams, the thread is tied to every peg. In some, it is not. Is it related to the two numbers (the number of pegs and the thread-gap) being co-prime?

Make such pictures for the following:

a. 15 pegs, thread-gap of 10

b. 10 pegs, thread-gap of 7

c. 14 pegs, thread-gap of 6

d. 8 pegs, thread-gap of 3

Below are some boxes with four numbers in each box. Within each box try to say how each number is special compared to the rest. Share with your classmates and find out who else gave the same reasons as you did. Did anyone give different reasons that may not have occurred to you?

A prime puzzle

The figure on the left shows the puzzle. The figure on the right shows the solution of the puzzle. Think what the rules can be to solve the puzzle.

Rules

Fill the grid with prime numbers only so that the product of each row is the number to the right of the row and the product of each column is the number below the column.

Q. Solve the prime puzzles by filling the grids with prime numbers only so that the product of each row is the number to the right of the row and the product of each column is the number below the column.

At what number is 'idli-vada' said for the 10th time?

If the game is played for the numbers 1 to 90, find out:

a. How many times would the children say 'idli' (including the times they say 'idli-vada')? b. How many times would the children say 'vada' (including the times they say 'idli-vada')? c. How many times would the children say 'idli-vada'?

What if the game was played till 900? How would your answers change?

Is this figure somehow related to the 'idli-vada' game?

Hint: Imagine playing the game till 30. Draw the figure if the game is played till 60.

Find all multiples of 40 that lie between 310 and 410.

Who am I?

a. I am a number less than 40. One of my factors is 7. The sum of my digits is 8.

b. I am a number less than 100. Two of my factors are 3 and 5. One of my digits is 1 more than the other.

A number for which the sum of all its factors is equal to twice the number is called a perfect number. The number 28 is a perfect number. Its factors are 1, 2, 4, 7, 14 and 28. Their sum is 56 which is twice 28. Find a perfect number between 1 and 10.

Find the common factors of:

a. 20 and 28

b. 35 and 50

c. 4, 8 and 12

d. 5, 15 and 25

Find any three numbers that are multiples of 25 but not multiples of 50.

Anshu and his friends play the 'idli-vada' game with two numbers, which are both smaller than 10. The first time anybody says 'idli-vada' is after the number 50. What could the two numbers be which are assigned 'idli' and 'vada'?

In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?

In the diagram below, Guna has erased all the numbers except the common multiples. Find out what those numbers could be and fill in the missing numbers in the empty regions.

Find the smallest number that is a multiple of all the numbers from 1 to 10, except for 7.

Find the smallest number that is a multiple of all the numbers from 1 to 10.

We see that 2 is a prime and also an even number. Is there any other even prime?

Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference?

Are there an equal number of primes occurring in every row in the table on the previous page? Which decades have the least number of primes? Which have the most number of primes?

Which of the following numbers are prime: 23, 51, 37, 26?

Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.

The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100.

Find seven consecutive composite numbers between 1 and 100.

Twin primes are pairs of primes having a difference of 2. For example, 3 and 5 are twin primes. So are 17 and 19. Find the other twin primes between 1 and 100.

Identify whether each statement is true or false. Explain.

a. There is no prime number whose units digit is 4.

b. A product of primes can also be prime.

c. Prime numbers do not have any factors.

d. All even numbers are composite numbers.

e. 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.

Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?

How many three-digit prime numbers can you make using each of 2, 4 and 5 once?

Observe that 3 is a prime number, and $2 \times 3 + 1 = 7$ is also a prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples.

Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000.

The prime factorisation of a number has one 2, two 3s, and one 11. What is the number?

Find three prime numbers, all less than 30, whose product is 1955.

Find the prime factorisation of these numbers without multiplying first

a. $56 \times 25$

b. $108 \times 75$

c. $1000 \times 81$

What is the smallest number whose prime factorisation has:

a. three different prime numbers?

b. four different prime numbers?

Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer.

a. 30 and 45

b. 57 and 85

c. 121 and 1331

d. 343 and 216

Is the first number divisible by the second? Use prime factorisation.

a. 225 and 27

b. 96 and 24

c. 343 and 17

d. 999 and 99

The first number has prime factorisation $2 \times 3 \times 7$ and the second number has prime factorisation $3 \times 7 \times 11$. Are they co-prime? Does one of them divide the other?

Guna says, “Any two prime numbers are co-prime?”. Is he right?

2024 is a leap year (as February has 29 days). Leap years occur in the years that are multiples of 4, except for those years that are evenly divisible by 100 but not 400.

a. From the year you were born till now, which years were leap years? b. From the year 2024 till 2099, how many leap years are there?

Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.

Explore and find out if each statement is always true, sometimes true or never true. You can give examples to support your reasoning.

a. Sum of two even numbers gives a multiple of 4. b. Sum of two odd numbers gives a multiple of 4.

Find the remainders obtained when each of the following numbers are divided by (a) 10, (b) 5, (c) 2.

78, 99, 173, 572, 980, 1111, 2345

The teacher asked if 14560 is divisible by all of 2, 4, 5, 8 and 10. Guna checked for divisibility of 14560 by only two of these numbers and then declared that it was also divisible by all of them. What could those two numbers be?

Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160.

Write two numbers whose product is 10000. The two numbers should not have 0 as the units digit.

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,

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?

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

1. Guna can arrange 12 figs in a rectangular manner.
2. 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?

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

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

a. 18 and 35

b. 15 and 37

c. 30 and 415

d. 17 and 69

e. 81 and 18

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?

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

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

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

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

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

Can you now answer if 8560 is divisible by 10?

Consider this statement:

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

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

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

Consider this statement:

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

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?

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

Is 8560 divisible by 2? Why or why not?

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?

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?

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.

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?

Is 8536 divisible by 4?

Consider these statements:

1. Only the last two digits matter when deciding if a given number is divisible by 4.
2. If the number formed by the last two digits is divisible by 4, then the original number is divisible by 4.
3. 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?

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?

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

Consider these statements:

1. Only the last three digits matter when deciding if a given number is divisible by 8.
2. If the number formed by the last three digits is divisible by 8, then the original number is divisible by 8.
3. 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?

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