Prime Time — Class 6 Mathematics
Ganita Prakash · Chapter 5
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5- Mathematics
- Science
- + 3 more
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
Chapter summary
Key ideas & terms
- Factor
- A number that divides another number exactly, leaving no remainder; e.g. 4 is a factor of 12.
- Multiple
- The product of a number with a counting number; e.g. 12, 24, 36 are multiples of 12.
- Prime number
- A number with exactly two factors, 1 and itself, such as
. - Composite number
- A number with more than two factors, such as
. - Co-prime numbers
- Two numbers whose only common factor is 1, such as 4 and 9.
- Prime factorisation
- Writing a number as a product of prime numbers, e.g.
.
Extra questions & answers
What is a prime number?
A number that has exactly two factors, namely 1 and the number itself, such as
What is a composite number?
A number with more than two factors, such as
Is 1 a prime or a composite number?
Neither; 1 has only one factor and is neither prime nor composite.
What is the only even prime number?
2 is the only even prime number; every other even number has 2 as an extra factor.
What does it mean for two numbers to be co-prime?
They are co-prime if their only common factor is 1, for example 4 and 9.
What is the prime factorisation of 56?
What is the Sieve of Eratosthenes?
A method of finding primes by crossing out multiples of each prime in turn, leaving the prime numbers circled.
How can prime factorisation tell if two numbers are co-prime?
If their prime factorisations share no common prime factor, the numbers are co-prime; otherwise they are not.
Numbers divisible by 5 end in which digits?
Numbers divisible by 5 end with either 0 or 5.
How do you test if a number is divisible by 4?
Check whether the number formed by its last two digits is divisible by 4; if so, the whole number is divisible by 4.
Figure it Out
1. At what number is ‘idli-vada’ said for the 10th time?
The numbers for ‘idli-vada’ are common multiples of
2. 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’)?
Multiples of
2. If the game is played for the numbers 1 to 90, find out: b. How many times would the children say ‘vada’ (including the times they say ‘idli-vada’)?
Multiples of
2. If the game is played for the numbers 1 to 90, find out: c. How many times would the children say ‘idli-vada’?
‘idli-vada’ are common multiples of
3. What if the game was played till 900? How would your answers change?
Up to
4. 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.
Yes. The Venn diagram in Fig. 5.1 shows multiples of
Let us now play the ‘idli-vada’ game with different pairs of numbers:
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. a. 2 and 5
For
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. b. 3 and 7
For
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. c. 4 and 6
For
Which of the following could be the other number:
Which of the following could be the other number: 2, 3, 5, 8, 10?
One number is
What jump size can reach both 15 and 30? There are multiple jump sizes possible. Try to find them all.
What jump size can reach both 15 and 30? There are multiple jump sizes possible. Try to find them all.
A jump size must be a common factor of
Look at the table below. What do you notice?
Look at the table below. What do you notice? 1. Is there anything common among the shaded numbers?
The shaded numbers are the multiples of 3 and the circled numbers are the multiples of 4. So every shaded number has 3 as a factor.
Look at the table below. What do you notice? 2. Is there anything common among the circled numbers?
The circled numbers are
Look at the table below. What do you notice? 3. Which numbers are both shaded and circled? What are these numbers called?
The numbers that are both shaded (multiples of 3) and circled (multiples of 4) are the common multiples of 3 and 4 — that is, the multiples of
Figure it Out
1. Find all multiples of 40 that lie between 310 and 410.
Multiples of
2. Who am I? a. I am a number less than 40. One of my factors is 7. The sum of my digits is 8.
Multiples of
2. Who am I? 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.
Having factors
3. 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.
Check
4. Find the common factors of: a. 20 and 28
Factors of
4. Find the common factors of: b. 35 and 50
Factors of
4. Find the common factors of: c. 4, 8 and 12
Factors of
4. Find the common factors of: d. 5, 15 and 25
Factors of
5. Find any three numbers that are multiples of 25 but not multiples of 50.
Multiples of
6. 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’?
We need two numbers
7. In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?
Need common factors of
8. 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. Multiples of ____ Common multiples 72 48 24 Multiples of ____
The common multiples shown are
9. Find the smallest number that is a multiple of all the numbers from 1 to 10, except for 7.
We need
10. Find the smallest number that is a multiple of all the numbers from 1 to 10.
Include
Figure it Out
1. We see that 2 is a prime and also an even number. Is there any other even prime?
No. Every even number greater than
2. Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference?
Primes up to
3. 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?
No, the distribution is not equal. Counting by tens:
4. Which of the following numbers are prime: 23, 51, 37, 26?
5. Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.
Primes
6. 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.
Such emirp pairs with same digits reversed are:
7. Find seven consecutive composite numbers between 1 and 100.
One block is
8. 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.
Twin prime pairs
9. Identify whether each statement is true or false. Explain. a. There is no prime number whose units digit is 4.
True. Any number ending in
9. Identify whether each statement is true or false. Explain. b. A product of primes can also be prime.
False. If
9. Identify whether each statement is true or false. Explain. c. Prime numbers do not have any factors.
False. Every prime has exactly two factors:
9. Identify whether each statement is true or false. Explain. d. All even numbers are composite numbers.
False.
9. Identify whether each statement is true or false. Explain. e. 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.
True. Every prime
10. Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?
11. How many three-digit prime numbers can you make using each of 2, 4 and 5 once?
Any number formed with digits
12. Observe that 3 is a prime number, and 2 × 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.
We need primes
Check if these pairs are safe:
Check if these pairs are safe: a. 15 and 39
Factors of
Check if these pairs are safe: b. 4 and 15
Factors of
Check if these pairs are safe: c. 18 and 29
Factors of
Check if these pairs are safe: d. 20 and 55
Factors of
Which of the following pairs of numbers are co-prime?
Which of the following pairs of numbers are co-prime? a. 18 and 35
Which of the following pairs of numbers are co-prime? b. 15 and 37
Which of the following pairs of numbers are co-prime? c. 30 and 415
Which of the following pairs of numbers are co-prime? d. 17 and 69
Which of the following pairs of numbers are co-prime? e. 81 and 18
While playing the ‘idli-vada’ game with different number pairs, Anshu observed something interesting!
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.
This happens when the two numbers are co-prime. Example:
While playing the ‘idli-vada’ game with different number pairs, Anshu observed something interesting! 2. At other times the first common multiple was less than the product of the two numbers.
This happens when the numbers are not co-prime and share a factor. Example:
Figure it Out
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 64
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 104
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 105
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 243
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 320
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 141
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 1728
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 729
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 1024
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 1331
1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 1000
2. The prime factorisation of a number has one 2, two 3s, and one 11. What is the number?
The number is
3. Find three prime numbers, all less than 30, whose product is 1955.
4. Find the prime factorisation of these numbers without multiplying first a. 56 × 25
4. Find the prime factorisation of these numbers without multiplying first b. 108 × 75
4. Find the prime factorisation of these numbers without multiplying first c. 1000 × 81
5. What is the smallest number whose prime factorisation has: a. three different prime numbers?
Use the three smallest primes
5. What is the smallest number whose prime factorisation has: b. four different prime numbers?
Use the four smallest primes
Figure it Out
1. Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. a. 30 and 45
1. Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. b. 57 and 85
1. Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. c. 121 and 1331
1. Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. d. 343 and 216
2. Is the first number divisible by the second? Use prime factorisation. a. 225 and 27
2. Is the first number divisible by the second? Use prime factorisation. b. 96 and 24
2. Is the first number divisible by the second? Use prime factorisation. c. 343 and 17
2. Is the first number divisible by the second? Use prime factorisation. d. 999 and 99
3. The first number has prime factorisation 2 × 3 × 7 and the second number has prime factorisation 3 × 7 × 11. Are they co-prime? Does one of them divide the other?
First
4. Guna says, “Any two prime numbers are co-prime?”. Is he right?
Yes. Two distinct primes
Figure it Out
1. 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?
Leap years are numbers divisible by
1. 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. b. From the year 2024 till 2099, how many leap years are there?
From
2. Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.
A palindrome reads the same forwards and backwards (form
3. 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.
Sometimes true.
3. Explore and find out if each statement is always true, sometimes true or never true. You can give examples to support your reasoning. b. Sum of two odd numbers gives a multiple of 4.
Sometimes true.
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 (a) 10
Remainder on dividing by
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 (b) 5
Remainder mod
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 (c) 2
Remainder mod
5. 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?
If a number is divisible by
6. Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160.
A number divisible by all of