Extra Questions Answers for Class 6 Maths Chapter 5 Prime Time - Ganita Prakash
Chapter 5 Prime Time Extra Questions Answers for Class 6 Maths is provided here by studyrankers. All the questions are crafted by our experts by keeping in mind that all the important points must be covered. You can also Download PDF of Class 6 Maths Chapter 5 Prime Time Extra Questions which will boost the student confidence and help in solving the exercises and questions from the chapter. The chapter is taken from the new NCERT Mathematics textbook, Ganita Prakash.
These Revision Notes for Class 6 Maths will develop you understanding of the chapter and help in gaining good marks in the examinations. We have also provided Chapter 5 Prime Time NCERT Solutions which will help you in completing your homework on time. These NCERT Solutions will help an individual to increase concentration and you can solve questions of supplementary books easily. Students can also check Revision Notes for Prime Time Class 6 Maths to prepare for their examination completely.
Important Questions for Chapter 5 Prime Time Class 6 Maths
Multiple Choice Questions
Question 1. Which of the following is a prime number?
(a) 49
(b) 51
(c) 53
(d) 55
Answer
(c) 53
53 is a prime number as it has no divisors other than 1 and 53.
Question 2. What is the first common multiple of 3 and 5?
(a) 10
(b) 12
(c) 15
(d) 20
Answer
(c) 15
The first common multiple of 3 and 5 is 15 because it is the smallest number that both 3 and 5 can divide into without a remainder.
Question 3. How many prime numbers are there between 1 and 10?
(a) 2
(b) 3
(c) 4
(d) 5
Answer
(c) 4
The prime numbers between 1 and 10 are 2, 3, 5, and 7, making a total of 4.
Question 4. Which pair of numbers is co-prime?
(a) 12 and 18
(b) 14 and 21
(c) 8 and 9
(d) 10 and 20
Answer
(c) 8 and 9
8 and 9 are co-prime because they have no common factors other than 1.
Question 5. The smallest number that is a multiple of both 3 and 4 is:
(a) 6
(b) 9
(c) 12
(d) 15
Answer
(c) 12
12 is the smallest number that is a multiple of both 3 and 4 because it is the lowest common multiple of these two numbers.
Fill in the Blanks
Question 1. The smallest prime number is ____.
Answer
2
The smallest prime number is 2, which is also the only even prime number. All other even numbers are composite.
Question 2. Numbers that have only two factors, 1 and the number itself, are called ____.
Answer
Prime numbers
A prime number is a number that can only be divided evenly by 1 and itself, such as 3, 5, 7, etc.
Question 3. The common multiples of 3 and 5 within the first 100 numbers are ____, ____, and ____.
Answer
15, 30, 45
A common multiple of two numbers is a number that is a multiple of both. For 3 and 5, the common multiples include 15, 30, and 45.
Question 4. The Sieve of ____ is a method used to find all prime numbers up to a certain number.
Answer
Eratosthenes
The Sieve of Eratosthenes is an ancient algorithm used to find all primes up to a specified integer by progressively marking the multiples of each prime starting from 2.
Question 5. Numbers that are neither prime nor composite are ____.
Answer
1
The number 1 is unique because it only has one factor, itself, and is thus neither prime nor composite.
Question 6. The numbers which have more than two factors are called ____.
Answer
Composite
A composite number has more than two factors. For example, 4 is a composite number because it has factors 1, 2, and 4.
Question 7. The numbers which are not multiples of 2 are known as _____.
Answer
Odd
Odd numbers cannot be divided evenly by 2. For example, 3 and 7 are odd numbers because they don’t divide by 2 without leaving a remainder.
Question 8. The two numbers which have only 1 as their common factor are called _____.
Answer
Co-primes
Co-prime numbers have no common factors other than 1. For instance, 8 and 15 are co-prime because 1 is their only common factor.
Question 9. The number which is neither prime nor composite is _____.
Answer
1
The number 1 is special because it has only one factor, which is itself. It doesn’t meet the criteria to be classified as either a prime number (which has exactly two factors) or a composite number (which has more than two factors).
Question 10. Every number is a ____ and ____ of itself.
Answer
Factor, Multiple
Every number can divide itself exactly (making it a factor of itself) and can also be multiplied by 1 to get itself (making it a multiple of itself). For example, 5 is both a factor and a multiple of 5.
True or False
Question 1. 9 is a prime number.
Answer
False
9 is not a prime number because it can be divided by 1, 3, and 9.
Question 2. The sum of two odd numbers and one even number is even.
Answer
True
3 + 5 + 6 = 14, i.e., even
Question 3. The product of three odd numbers is odd.
Answer
True
3×5×7 = 105, i.e., odd
Question 4. All prime numbers are odd.
Answer
False
2 is a prime number and it is also even
Question 5. Prime numbers do not have any factors.
Answer
False
1 and the number itself are factors of the number
Question 6. The sum of three odd numbers is even.
Answer
False
3 + 5 + 7 = 15, i.e., odd
Question 7. 2 is the only even prime number.
Answer
True
Question 8. All even numbers are composite numbers.
Answer
False
2 is a prime number
Question 9. The number 2 is the only even prime number.
Answer
True
2 is the only even prime number because it can only be divided evenly by 1 and 2.
Question 10. The sum of two prime numbers is always even.
Answer
False
2 + 3 = 5, i.e., odd
Question 11. All multiples of 4 are also multiples of 2.
Answer
True
Every multiple of 4 is also a multiple of 2 because 4 is divisible by 2.
Question 12. The product of two even numbers is always even.
Answer
True
2×4 = 8, i.e., even
Question 13. If a number is divisible by 8, it is also divisible by 4.
Answer
True
Since 8 is divisible by 4, any number divisible by 8 will also be divisible by 4.
Question 14. If an even number is divided by 2, the quotient is always odd.
Answer
False
4 ÷ 2 = 2, i.e., even
Question 15. There is no prime number whose units digit is 4. (T/F)
Answer
True
A prime number must end in 1, 3, 7, or 9 (except for the number 2). Any number ending in 0, 2, 4, 6, or 8 is divisible by 2, making it a non-prime.
Question 16. A product of primes can also be prime. (T/F)
Answer
False
A product of prime numbers is only prime if it involves precisely one prime number. When two or more prime numbers are multiplied, the result is a composite number, not a prime.
Question 17. The number 37 is a composite number.
Answer
False
37 is a prime number as it only has two factors: 1 and 37.
One Word Answer
Question 1. List all the prime numbers between 10 and 20.
Answer
11, 13, 17, 19
These numbers cannot be divided by any number other than 1 and themselves, making them prime.
Question 2. Find the common factors of 24 and 36.
Answer
1, 2, 3, 4, 6, 12
The common factors of 24 and 36 are the numbers that can divide both 24 and 36 without leaving a remainder.
Question 3. What is the prime factorization of 72?
Answer
2 × 2 × 2 × 3 × 3
The prime factorization of 72 involves breaking it down into its prime factors.
Question 4. Identify two numbers between 1 and 50 that are co-prime.
Answer
14 and 25
14 and 25 are co-prime because they have no common factors other than 1.
Question 5. What is the smallest multiple of 5 that is also a multiple of 3?
Answer
15
15 is the smallest number that is a multiple of both 5 and 3.
Solve this:
Question 1. Is the first number divisible by the second? Use prime factorisation.
a. 150 and 25
b. 84 and 12Solution
a. 150 and 25
Yes, 150 is divisible by 25.
Explanation: The prime factorization of 150 is 2 × 3 × 5 × 5, and the prime factorization of 25 is 5 × 5. Since 150 has all the factors of 25, it is divisible by 25.
b. 84 and 12
Yes, 84 is divisible by 12.
Explanation: The prime factorization of 84 is 2 × 2 × 3 × 7, and the prime factorization of 12 is 2 × 2 × 3. Since 84 has all the factors of 12, it is divisible by 12.
Question 2. Find three prime numbers, all less than 50, whose product is 2310.
Solution
The prime factorization of 2310:
2310 = 2 × 3 × 5 × 7 × 11
Among these, select three prime numbers that are less than 50: 2, 3, and 5.
Hence, the three prime numbers whose product is 2310 are 2, 3, and 5.
Question 3. What is the smallest number whose prime factorization has:
a. Three different prime numbers?b. Four different prime numbers?
Solution
a. The smallest prime numbers are 3, 5, and 7. To find the smallest number with these primes as factors, multiply them together:
3 × 5 × 7 = 105
So, the smallest number whose prime factorization has three different prime numbers is 105.
b. The smallest four prime numbers are 3, 5, 7, and 11. To find the smallest number with these primes as factors, multiply them together:
3 × 5 × 7 × 11 = 1155
Thus, the smallest number whose prime factorization has four different prime numbers is 1155.
Question 4. Who am I?
I am a number less than 50. One of my factors is 6. The sum of my digits is 9.
Solution
6 is a factor of 6, 12, 18, 24, 30, 36, 42, 48, which are less than 50. Among these, the number whose digit sum is 9 is 36 (3 + 6 = 9).
So, I am 36.
Explanation: The number is 36 (since 3 + 6 = 9 and 36 is divisible by 6).
Question 5. Which of the following numbers is the product of exactly three distinct prime numbers: 50, 84, 105, 280?
Solution
Here,
50 = 2 × 5 × 5 (2 distinct primes)
84 = 2 × 2 × 3 × 7 (4 distinct primes)
105 = 3 × 5 × 7 (3 distinct primes)
280 = 2 × 2 × 2 × 5 × 7 (4 distinct primes)
Number 105 is the product of exactly three distinct prime numbers i.e. 3 × 5 × 7.
Question 6. Answer the following Questions.
(i) Find all the multiples of 13 up to 100.
Answer
13, 26, 39, 52, 65, 78, 91
(ii) Write all the factors of 120.
Answer
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
(iii) Identify the numbers below which are multiples of 45.
270, 295, 305, 315, 333, 360, 400
Answer
270, 315, 360
(iv) 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.
Answer
17, 71
37, 73
79, 97
(v) Write down separately the prime and composite numbers less than 20.
Answer
Prime numbers less than 20 are
2, 3, 5, 7, 11, 13, 17, 19
Composite numbers less than 20 are
4, 6, 8, 9, 10, 12, 14, 15, 16, 18
(vi) What is the greatest prime number between 1 and 10?
Answer
Prime numbers between 1 and 10 are 2, 3, 5, and 7. Among these numbers, 7 is the greatest.
(vii) Express the following as the sum of two odd primes.
(a) 44
(b) 36
(c) 24
(d) 18
Answer
(a) 44 = 37 + 7
(b) 36 = 31 + 5
(c) 24 = 19 + 5
(d) 18 = 11 + 7
(viii) Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
Answer
Between 89 and 97, both of which are prime numbers, there are 7 composite numbers. They are
90, 91, 92, 93, 94, 95, 96
Numbers Factors:
- 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
- 91 = 1, 7, 13, 91
- 92 = 1, 2, 4, 23, 46, 92
- 93 = 1, 3, 31, 93
- 94 = 1, 2, 47, 94
- 95 = 1, 5, 19, 95
- 96 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
(ix) Write a digit in the blank space of each of the following numbers so that the number formed is divisible by 11:
(a) 92 ___ 389
(b) 8 ___9484
Answer
(a) 92____389
Let a be placed in the blank.
Sum of the digits at odd places = 9 + 3 + 2 = 14
Sum of the digits at even places = 8 + a + 9 = 17 + a
Difference = 17 + a − 14 = 3 + a
For a number to be divisible by 11, this difference should be zero or a multiple of 11.
If 3 + a = 0, then
a = − 3
However, it cannot be negative.
The closest multiple of 11, which is near to 3, has to be taken. It is 11 itself.
3 + a = 11
a = 8
Therefore, the required digit is 8.
(b) 8____9484
Let a be placed in the blank.
Sum of the digits at odd places = 4 + 4 + a = 8 + a
Sum of the digits at even places = 8 + 9 + 8 = 25
Difference = 25 − (8 + a)
= 17 − a
For a number to be divisible by 11, this difference should be zero or a multiple of 11.
If 17 − a = 0, then
a = 17
This is not possible.
A multiple of 11 has to be taken. Taking 11, we obtain
17 − a = 11
a = 6
Therefore, the required digit is 6.
(x) A number is divisible by both 5 and 12. By which another number will that number be always divisible?
Answer
Factors of 5 = 1, 5
Factors of 12 = 1, 2, 3, 4, 6, 12
As the common factor of these numbers is 1, the given two numbers are co-prime and the number will also be divisible by their product, i.e. 60, and the factors of 60, i.e., 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
(xi) A number is divisible by 12. By what other number will that number be divisible?
Answer
Since the number is divisible by 12, it will also be divisible by its factors i.e., 1, 2, 3, 4, 6, 12. Clearly, 1, 2, 3, 4, and 6 are numbers other than 12 by which this number is also divisible.
(xii) The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.
Answer
2×3×4 = 24, which is divisible by 6
9×10×11 = 990, which is divisible by 6
20×21×22 = 9240, which is divisible by 6
(xiii) The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.
Answer
3 + 5 = 8, which is divisible by 4
15 + 17 = 32, which is divisible by 4
19 + 21 = 40, which is divisible by 4
(xiv) Determine if 25110 is divisible by 45.
[Hint: 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9].
Answer
45 = 5×9
Factors of 5 = 1, 5
Factors of 9 = 1, 3, 9
Therefore, 5 and 9 are co-prime numbers.
Since the last digit of 25110 is 0, it is divisible by 5.
Sum of the digits of 25110 = 2 + 5 + 1 + 1 + 0 = 9
As the sum of the digits of 25110 is divisible by 9, therefore, 25110 is divisible by 9.
Since the number is divisible by 5 and 9 both, it is divisible by 45.
(xv) Find the LCM of 12, 16, 24 and 36?
Answer
12 = 2×2×3
16 = 2×2×2×2
24 = 2×2×2×3
36 = 2×2×3×3
LCM = 2×2×2×2×3×3 = 144
(xvi) Find the HCF of 70, 105, 175
Answer
70 = 2×5×7
105 = 3×7×5
175 = 5×7×7
HCF = 5×7 = 35
(xvii) You have 24 small bottles of lemonade and 36 large bottles. What is the largest number of equal-sized groups you can make if each group has the same number of small and large bottles?
Answer
24 small bottles and 36 large bottles
HCF of 24 and 36 = 12
Largest number of equal-sized groups = 12
Question 7. Find all multiples of 50 that lie between 500 and 600.
Answer
Here, multiples of 50 are: 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600
Hence, multiples of 50 that lie between 500 and 600 are: 550.
Question 8. How many prime numbers are there from 10 to 20?
Answer
In total, there are 4 prime numbers between 10 and 20.
They are 11, 13, 17, and 19.
Question 9. How many composite numbers are there from 10 to 20?
Answer
The total number of composite numbers from 10 to 20 is 6.
They are 10, 12, 14, 15, 16, and 18.
Question 10. Is the first number divisible by the second? Use prime factorization.
(a) 150 and 25
(b) 84 and 28
(c) 224 and 16
(d) 800 and 80
Answer
(a) Prime Factors of 150 and 25:
150 = 2 × 3 × 5 × 5, and 25 = 5 × 5
Since 150 contains sufficient factors of 5, it is divisible by 25.
(b) Prime Factors of 84 and 28:
84 = 2 × 2 × 3 × 7, and 28 = 2 × 2 × 7
Since 84 includes the required factors to match those in 28, it is divisible by 28.
(c) Prime Factors of 224 and 16:
224 = 2 × 2 × 2 × 2 × 7, and 16 = 2 × 2 × 2 × 2
Since 224 includes the required factors to match those in 16, it is divisible by 16.
(d) Prime Factors of 800 and 80:
800 = 2 × 2 × 2 × 2 × 5 × 5, and 80 = 2 × 2 × 2 × 2 × 5
Since 800 includes the required factors to match those in 80, it is divisible by 80.
Question 11. Observe that 5 is a prime number, and 2 × 5 + 1 = 11 is also a prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples.
Answer
The five prime numbers for which doubling and adding 1 gives another prime are:
- 2 (since 2 × 2 + 1 = 5)
- 3 (since 2 × 3 + 1 = 7)
- 7 (since 2 × 7 + 1 = 15, which is not prime)
- 13 (since 2 × 13 + 1 = 27, which is not prime)
- 17 (since 2 × 17 + 1 = 35, which is not prime)
Question 12. Find the prime factorization of these numbers without multiplying first.
(a) 72 × 36
(b) 120 × 48
Answer
(a) Prime factors of 72 = 2 × 2 × 2 × 3 × 3
Prime factors of 36 = 2 × 2 × 3 × 3
Combined prime factorization of 72 × 36 = 2 × 2 × 2 × 3 × 3 × 2 × 2 × 3 × 3
(b) Prime factors of 120 = 2 × 2 × 2 × 3 × 5
Prime factors of 48 = 2 × 2 × 2 × 2 × 3
Combined prime factorization of 120 × 48 = 2 × 2 × 2 × 3 × 5 × 2 × 2 × 2 × 2 × 3
Question 13. Which of the following pairs of numbers are co-prime?
(a) 24 and 35
(b) 40 and 97
(c) 50 and 225
Answer
(a) Here, factors of 24 = 1 × 2 × 2 × 2 × 3, and factors of 35 = 1 × 5 × 7
No common factor other than 1.
Hence, 24 and 35 are co-prime numbers.
(b) We have factors of 40 = 1 × 2 × 2 × 2 × 5, and factors of 97 = 1 × 97
No common factor other than 1.
Hence, 40 and 97 are co-prime numbers.
(c) Given numbers are 50 and 225.
Here, factors of 50 = 1 × 2 × 5 × 5, and factors of 225 = 1 × 3 × 3 × 5 × 5.
Clearly, 5 is a common factor of 50 and 225.
Hence, 50 and 225 are not co-prime numbers.
Question 14. Consider these statements:
(a) Only the last two digits matter when deciding if a given number is divisible by 6.
(b) If the number formed by the last two digits is divisible by 6, then the original number is divisible by 6.
(c) If the original number is divisible by 6, then the number formed by the last two digits is divisible by 6.
Do you agree? Why or why not?
Answer
(a) Yes, that's correct. When determining if a number is divisible by 6, the divisibility rules for 6 focus on whether the number formed by the last two digits is divisible by 6, as 100 is divisible by 6.
(b) No, this is not correct. If the number formed by the last two digits is divisible by 6, it does not necessarily mean the entire number is divisible by 6. For example, 156 is divisible by 6, but 56 is not.
(c) Yes, that's correct. If the original number is divisible by 6, the number formed by the last two digits is also divisible by 6.
Question 15. Who am I?
(a) I am a number less than 50. One of my factors is 8. The sum of my digits is 10.
(b) I am a number less than 100. Two of my factors are 4 and 6. One of my digits is twice the other.
Answer
(a) 8 is a common factor of 8, 16, 24, 32, 40, and 48, which are less than 50. The number with a digit sum of 10 is 37.
So, I am 37.
(b) Common factors of 4 and 6 are 12, 24, 36, 48, 60, and 72 (which are less than 100). The number where one digit is twice the other is 36. So, I am 36.