QUESTION BANK Chapter 5: Prime Time
Multiple Choice Questions (1 Mark Each)
1. Which of the following is a prime number?
a) 21 b) 33 c) 47 d) 51 (Competency: Identifying prime numbers)
2. The first number for which a player says 'idli-vada' in the game is:
a) 10 b) 15 c) 20 d) 30 (Competency: Finding common multiples)
3. Which pair of numbers is co-prime?
a) 18 and 35 b) 15 and 39 c) 30 and 415 d) 81 and 18 (Competency: Identifying co-prime numbers)
4. The prime factorisation of 84 is:
a) 2 x 2 x 21 b) 2 x 3 x 14 c) 2 x 2 x 3 x 7 d) 3 x 4 x 7 (Competency: Prime factorisation)
5. A number is divisible by 8 if:
a) Its last digit is even. b) The number formed by its last two digits is divisible by 4.
c) The number formed by its last three digits is divisible by 8. d) It is divisible by both 2 and 4.
(Competency: Applying divisibility rules)
6. Which of the following is a perfect number?
a) 10 b) 12 c) 6 d) 8 (Competency: Understanding properties of numbers)
7. The largest 2-digit prime number is:
a) 91 b) 93 c) 97 d) 99 (Competency: Identifying prime numbers)
8. A number is divisible by 5 if:
a) It is an odd number. b) Its last digit is 0 or 5. c) The sum of its digits is 5. d) It ends with 0. (Competency: Applying divisibility rules)
9. Twin primes are pairs of primes that differ by:
a) 1 b) 2 c) 3 d) 4 (Competency: Understanding properties of primes)
10. The sum of the first five prime numbers is:
a) 18 b) 26 c) 28 d) 30 (Competency: Calculation and properties of primes)
11. The number 1 is:
a) A prime number b) A composite number c) Neither prime nor composite d) An even number
(Competency: Classifying numbers)
12. Which of the following numbers is divisible by 6?
a) 2341 b) 5732 c) 8460 d) 2953 (Competency: Applying divisibility rules)
13.15. The prime factorisation of 1000 is:
a) 10³ b) 2³ x 5³ c) 2² x 5³ d) 2³ x 5² (Competency: Prime factorisation)
14. A number is divisible by 9 if:
a) It is divisible by 3. b) Its last digit is 9. c) The sum of its digits is divisible by 9.
d) It is an odd number. (Competency: Applying divisibility rules)
15. Which of the following is a composite number?
a) 17 b) 23 c) 31 d) 39 (Competency: Identifying composite numbers)
16. The smallest 4-digit number divisible by 3 is:
a) 1000 b) 1002 c) 1005 d) 1010 (Competency: Applying divisibility rules)
17. The number of factors of a prime number is:
a) 1 b) 2 c) 3 d) Infinite (Competency: Understanding factors of primes)
18. The Sieve of Eratosthenes is used to find:
a) Multiples of a number b) Prime numbers c) Composite numbers d) Common factors
(Competency: Understanding mathematical methods)
19.The only even prime number is __________________
20. Which of the following numbers are prime? 23 ,51 , 37, 26
Assertion (A) and Reason (R) Type Questions (1 Mark Each)
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
a) Both A and R are true, and R is the correct explanation of A.
b) Both A and R are true, but R is NOT the correct explanation of A.
c) A is true, but R is false.
d) A is false, but R is true.
1. Assertion (A): 1 is neither a prime nor a composite number.
Reason (R): 1 has only one factor. (Competency: Classifying numbers)
2. Assertion (A): 15 and 32 are co-prime numbers.
Reason (R): Two numbers are co-prime if their Highest Common Factor is 1. (Competency: Understanding co-prime numbers)
3. Assertion (A): All even numbers are composite numbers.
Reason (R): 2 is a prime number. (Competency: Properties of even numbers and primes)
4. Assertion (A): The number 123456 is divisible by 3.
Reason (R): A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 123456 is 21, which is divisible by 3. (Competency: Applying divisibility rules)
5. Assertion (A): The prime factorisation of a number is unique.
Reason (R): The order of factors does not matter in multiplication. (Competency: Fundamental Theorem of Arithmetic)
6. Assertion (A): 91 is a prime number.
Reason (R): It has no factors other than 1 and itself. (Competency: Identifying prime numbers)
7. Assertion (A): The numbers 0 and 1 are not considered prime numbers.
Reason (R): A prime number must have exactly two distinct factors. (Competency: Definition of prime numbers)
8. Assertion (A): The number 1001 is divisible by 7.
Reason (R): The difference between the number formed by the last three digits and the remaining part (1 - 1 = 0) is divisible by 7. (Competency: Applying divisibility rules)
9. Assertion (A): Every composite number can be expressed as a product of primes.
Reason (R): This is known as the Fundamental Theorem of Arithmetic. (Competency: Fundamental Theorem of Arithmetic)
10. Assertion (A): 45, 60, 75, and 90 are all multiples of 15.
Reason (R): 15 is a common factor of all these numbers. (Competency: Understanding multiples and factors)
11. Assertion (A): The sum of two prime numbers is always even.
Reason (R): 2 is the only even prime number. (Competency: Properties of prime numbers)
12. Assertion (A): 864 is divisible by 8.
Reason (R): The number formed by its last three digits, 864, is divisible by 8.
(Competency: Applying divisibility rules)
13. Assertion (A): The number 1 is a unique number.
Reason (R): It is the multiplicative identity. (Competency: Properties of 1)
14. Assertion (A): 0 is a multiple of every number.
Reason (R): 0 divided by any number gives 0. (Competency: Understanding multiples)
15. Assertion (A): All numbers ending with 7 are prime numbers.
Reason (R): 27 is a composite number. (Competency: Identifying prime and composite numbers)
16. Assertion (A): The divisibility rule for 4 involves the last two digits.
Reason (R): 100 is divisible by 4. (Competency: Understanding divisibility rules)
17. Assertion (A): The number of prime numbers is finite.
Reason (R): Euclid proved that there are infinitely many primes. (Competency: Knowledge about primes)
True or False (1 Mark Each)
1. The number 57 is prime. (True/False) (Competency: Identifying prime numbers)
2. 1 is the smallest prime number. (True/False) (Competency: Identifying prime numbers)
3. Every multiple of a number is greater than or equal to that number. (True/False) (Competency: Understanding multiples)
4. The sum of two odd numbers is always even. (True/False) (Competency: Properties of odd and even numbers)
5. If a number is divisible by 10, it must be divisible by 5. (True/False) (Competency: Relating divisibility rules)
6. All prime numbers are odd. (True/False) (Competency: Properties of prime numbers)
7. Two different numbers can have the same prime factorisation. (True/False) (Competency: Fundamental Theorem of Arithmetic)
8. A number is divisible by 3 if its last digit is divisible by 3. (True/False) (Competency: Applying divisibility rules)
9. Co-prime numbers must be prime numbers. (True/False) (Competency: Understanding co-prime numbers)
10. 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.
Short Answer Type Questions-I (2 Marks Each)
1. List all the factors of 36. (Competency: Finding factors)
2. Find all multiples of 40 that lie between 310 and 410. (Competency: Finding multiples)
3. Find the common factors of 20 and 28. b) 35 and 50 c) 4,8 and 12 d) 5,15 and 25 (Competency: Finding common factors)
4. Write the prime factorisation of 98. (Competency: Prime factorisation)
5. Check whether 23456 is divisible by 4. Give a reason for your answer. (Competency: Applying divisibility rules)
6. Is 1,23,456 divisible by 3? Justify your answer. (Competency: Applying divisibility rules)
7. Find any three numbers that are multiples of 25 but not multiples of 50 . (Competency: Understanding multiples)
8. Write the smallest and largest 2-digit prime numbers. (Competency: Identifying prime numbers)
9. list of primes till 100What is the smallest difference between two successive primes? What is the largest difference?
10. If a number is divisible by 9, will it always be divisible by 3? Justify with an example. (Competency: Relating divisibility rules)
11. Find the first three common multiples of 3 and 4. (Competency: Finding common multiples)
12. State whether 31 and 44 are co-prime. Give reasons. (Competency: Identifying co-prime numbers)
13. Find the missing number: The prime factorisation of ______ is 2³ × 3 × 5. (Competency: Prime factorisation)
14. Find the sum of the first three composite numbers. (Competency: Identifying composite numbers)
15. Who am I? (a.) Less than 40, factor 7, sum of digits is 8.
(b.) Less than 100, factors 3 and 5, one digit is 1 more than the other.
16. Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.
17. 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.
18. a) Find the smallest number that is a multiple of all the numbers from 1 to 10 except for 7 b) Find the smallest number that is a multiple of all the numbers from 1 to 10
19. Find seven consecutive composite numbers between 1 and 100.
20. The numbers 13 and 31 are prime numbers.Both these numbers have the same digits 1 and 3. Find such pairs of prime numbers up to 100.
21. 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.
Short Answer Type Questions-II (3 Marks Each)
1. Find the prime factorisation of 1728. (Competency: Prime factorisation)
2. Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?
3. How many three-digit prime numbers can you make using each of 2, 4 and once?
4. 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. Who am I? (Find all possible numbers) (Competency: Problem-solving based on factors)
5. Explain the Sieve of Eratosthenes. Use it to find all prime numbers between 1 and 30. (Competency: Understanding and applying the Sieve)
6.23. observe that 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.
7. Check whether 12ⁿ can end with the digit 0 for any natural number n. Justify your answer. (Competency: Reasoning with prime factorisation)
8.Write two numbers whose product is 10000. The two numbers should not have 0 as the units digit
9. A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or divisible by 11. Check the divisibility of 10934 by 11. (Competency: Applying divisibility rule for 11)
11. 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’?
12. In the treasure hunting game, Grumpy has kept treasures on 28 and 70 What jump sizes will land on both the numbers?
13. Which of the following pairs of numbers are co-prime? a) 18 and 35 (b)30 and 415 (c ) 15 and 37
(d) 17 and 69 (e) 81 and 18
14. Find the prime factorisations of the following numbers: 64, 104, 105, 243,320, 141, 1728, 1024, 1331, 1000 15. The prime factorisation of a number has one 2, two 3s , and one 11. What is the number?
16. Find three prime numbers, all less than 30 , whose product is 1955.
17. Find the prime factorisation of these numbers without multiplying first a. 56 × 25 (b) 108 × 75 © 1000 × 81 18. What is the smallest number whose prime factorisation has: (a) three different prime numbers? (b) four different prime numbers?
19. 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.
20. 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.
21. 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?
22. Guna says, any two prime numbers are co-prime. Is he right?
23. Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.
24.Explore and find out if each statement is always true, sometimes true or never true ou 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
25. Find the remainders obtained when each of the following numbers are divided by i) 10, ii) 5 iii) 3
78,99,173,572, 980, 1111,2345
26.Which of the following numbers are divisible by all of 2, 4, and 10: 572, 2352, 5600, 6000, 77622160.
27. 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?
Long Answer Type Questions (5 Marks Each)
1. a) b) Find all prime numbers between 50 and 80 using the Sieve of Eratosthenes.
(Competency: Understanding, applying, and explaining concepts)
2) Find the prime factorisation of 1000 and 1728.
3. a) Explain the rules for divisibility by 2, 3, 4, 5, 8, 9, and 10 with examples.
b) Using these rules, check the divisibility of 12,345 by 3, 5, and 10. Check the divisibility of 10,248 by 4 and 8.
(Competency: Understanding and applying multiple divisibility rules)
4. Check whether 81 and 16 are co-prime.
5) Two numbers are co-prime and their product is 217. Find the numbers. (Competency: Understanding and problem-solving with co-prime numbers)
6. a) List all twin prime pairs under 50.
b) 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.
c) Find a perfect number between 1 and 10. or Find the smallest perfect number. (Competency: Understanding special types of numbers)
7. a) Find the smallest number which when increased by 17 is exactly divisible by 28, 36, and 45.
8. a) Is 7 x 11 x 13 + 13 a composite number? Justify your answer.
b) Check whether 15ⁿ can end with the digit zero for any natural number n.
c) Find the digit ‘a’ if 37a5 is divisible by 3. (Competency: Reasoning and justification using number theory)
9. a) Draw a factor tree for 240. b) Write its prime factorisation.
c) Using this, find the number of factors of 240. (Hint: If N = a^p x b^q x c^r, number of factors = (p+1)(q+1)(r+1)) (Competency: Factor tree, prime factorisation, and finding number of factors)
10. 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?
11. In a game, children sit in a circle. They call out numbers from 1. For multiples of 3, they say 'Idli'. For multiples of 5, they say 'Vada'. For numbers which are multiples of both 3 and 5, they say 'Idli-Vada'.
1. At what number is ‘idli-vada’ said for the 10th time?
2. If the game is played for the numbers from 1 till 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’?
3. What if the game was played till 900? How would your answers change?
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
Case-Based Questions (4 MCQs each)
Case Study 1: The Idli-Vada Game
In a game, children sit in a circle. They call out numbers from 1. For multiples of 3, they say 'Idli'. For multiples of 5, they say 'Vada'. For numbers which are multiples of both 3 and 5, they say 'Idli-Vada'.
1. On which number will a child first say 'Idli-Vada'? a) 5 b) 10 c) 15 d) 30 (Competency: Finding common multiples)
2. Up to 30, how many times will 'Idli' be said?
a) 9 b) 10 c) 11 d) 12 (Competency: Counting multiples)
3. Up to 30, how many times will only 'Vada' be said (and not 'Idli-Vada')?
a) 3 b) 4 c) 5 d) 6 (Competency: Analyzing multiples)
4. If the game is played with rules for 4 ('Idli') and 6 ('Vada'), what is the first number for which a child will say 'Idli-Vada'? a) 12 b) 18 c) 24 d) 36 (Competency: Finding LCM)
Case Study 2: Jumpy's Treasure Hunt
Jumpy can only jump in steps of equal size, starting from 0. He wins a treasure if he lands exactly on the number where it is placed. Grumpy places treasures on two numbers: 18 and 24.
1. Which of the following jump sizes will land Jumpy on both 18 and 24?
a) 3 b) 4 c) 5 d) 7 (Competency: Finding common factors)
2. What is the largest jump size that can land him on both treasures?
a) 2 b) 3 c) 6 d) 8 (Competency: Finding HCF)
3. If Grumpy wants to make it impossible for Jumpy to win both treasures with any jump size greater than 1, which pair of numbers should he choose?
a) 12 and 18 b) 15 and 28 c) 16 and 24 d) 20 and 30 (Competency: Identifying co-prime numbers)
4. If a jump size lands Jumpy on number A, then A must be a ______ of the jump size.
a) factor b) multiple c) divisor d) prime (Competency: Understanding multiples and factors)
Case Study 3: Packing Figs
Guna and Anshu are packing figs. Guna wants to pack 12 figs per box, and Anshu wants to pack 7 figs per box. They are exploring the different rectangular arrangements possible for each number.
1. How many different rectangular arrangements can Guna make for 12 figs?
a) 2 b) 3 c) 4 d) 6 (Competency: Finding number of factor pairs)
2. How many different rectangular arrangements can Anshu make for 7 figs?
a) 0 b) 1 c) 2 d) 7 (Competency: Properties of prime numbers)
3. Based on this, 7 is a ______ number.
a) Prime b) Composite c) Co-prime d) Even (Competency: Classifying numbers)
4. Which of the following numbers will have the most number of rectangular arrangements?
a) 17 b) 23 c) 36 d) 41 (Competency: Relating to number of factors)
Case Study 4: The Sieve of Eratosthenes
The Sieve of Eratosthenes is an ancient algorithm used to find all prime numbers up to any given limit. It works by iteratively marking the multiples of each prime number starting from 2.
1. The first step in the Sieve is to circle the number __ and cross out all its multiples.
a) 1 b) 2 c) 3 d) 4 (Competency: Understanding the Sieve method)
2. After circling 2 and crossing out its multiples, the next number to be circled is:
a) 3 b) 4 c) 5 d) 6 (Competency: Applying the Sieve method)
3. The numbers crossed out as multiples of 2 and 3 (like 6, 12, 18) are:
a) Prime numbers b) Composite numbers c) Co-prime numbers d) Odd numbers (Competency: Identifying composite numbers)
4. How many prime numbers are there between 1 and 50?
a) 14 b) 15 c) 16 d) 17 (Competency: Applying the Sieve and counting)
Case Study 5: Leap Years
A leap year has 366 days, with February having 29 days. The rule for leap years is: If a year is divisible by 4, it is a leap year, except for end-of-century years, which must be divisible by 400 to be leap years.
1. Which of the following was a leap year?
a) 1900 b) 1978 c) 2000 d) 2022 (Competency: Applying divisibility rules in context)
2. The year 2024 is a leap year. What will be the next leap year?
a) 2025 b) 2026 c) 2027 d) 2028 (Competency: Finding multiples)
3. How many leap years are there between the years 2001 and 2100?
a) 23 b) 24 c) 25 d) 26 (Competency: Problem-solving with divisibility)
4. A century year (like 2100, 2200) will be a leap year only if it is divisible by:
a) 4 b) 40 c) 100 d) 400 (Competency: Remembering the exception rule)
