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NCERT Solutions · Maths Ch 5
Prime Time
55 questions · 5 exercises
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Ex 5.1
Ex 5.2
Ex 5.3
Ex 5.4
Ex 5.5
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
EXERCISE 5.1 · 7 QUESTIONS
Common Multiples
Q1
In the 'idli-vada' game (idli for multiples of 3, vada for multiples of 5), at what number is 'idli-vada' said for the 10th time?
(2 marks)
Q2
If the game is played for the numbers 1 to 90, find: (a) How many times would the children say 'idli' (including 'idli-vada')? (b) How many times 'vada' (including 'idli-vada')? (c) How many times 'idli-vada'?
(3 marks)
Q3
What if the game was played till 900? How would your answers change?
(2 marks)
Q4
Fig. 5.1 shows two overlapping circles, 'Multiples of 3' and 'Multiples of 5', with the numbers up to 30 placed in them. Is this figure related to the 'idli-vada' game? Draw the figure if the game is played till 60.
(3 marks)
Q5
Play the 'idli-vada' game with the pairs (a) 2 and 5, (b) 3 and 7, (c) 4 and 6. Draw a figure similar to Fig. 5.1 if the game is played up to 60.
(3 marks)
Q6
Yesterday we played the game with two numbers and ended up saying just 'idli' or 'idli-vada'; nobody said just 'vada'. One of the numbers was 4. Which of 2, 3, 5, 8, 10 could be the other number?
(3 marks)
Q7
What jump size can reach both 15 and 30? Find all.
(2 marks)
2
EXERCISE 5.2 · 10 QUESTIONS
Jump Jackpot
Q1
Find all multiples of 40 that lie between 310 and 410.
(2 marks)
Q2
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.
(3 marks)
Q3
A number for which the sum of all its factors is equal to twice the number is called a perfect number. Find a perfect number between 1 and 10.
(3 marks)
Q4
Find the common factors of: (a) 20 and 28 (b) 35 and 50 (c) 4, 8 and 12 (d) 5, 15 and 25.
(3 marks)
Q5
Find any three numbers that are multiples of 25 but not multiples of 50.
(2 marks)
Q6
Anshu and friends play the 'idli-vada' game with two numbers, both smaller than 10. The first time anybody says 'idli-vada' is after the number 50. What could the two numbers be?
(3 marks)
Q7
In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?
(2 marks)
Q8
In the diagram, Guna has erased all the numbers except the common multiples 24, 48 and 72 (shown in the overlap of 'Multiples of __' and 'Multiples of __'). Find what those numbers could be and fill in the missing numbers in the empty regions.
(2 marks)
Q9
Find the smallest number that is a multiple of all the numbers from 1 to 10, except for 7.
(3 marks)
Q10
Find the smallest number that is a multiple of all the numbers from 1 to 10.
(2 marks)
3
EXERCISE 5.3 · 13 QUESTIONS
Prime Numbers
Q1
Section 5.2 (page 113): How many prime numbers are there from 21 to 30? How many composite numbers are there from 21 to 30?
(3 marks)
Q2
We see that 2 is a prime and also an even number. Is there any other even prime?
(2 marks)
Q3
Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference?
(2 marks)
Q4
Are there an equal number of primes in every row (decade) of the table of 1 to 100? Which decades have the least number of primes? Which have the most?
(3 marks)
Q5
Which of the following numbers are prime: 23, 51, 37, 26?
(3 marks)
Q6
Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.
(2 marks)
Q7
The numbers 13 and 31 are primes with the same digits 1 and 3. Find such pairs of prime numbers up to 100.
(5 marks)
Q8
Find seven consecutive composite numbers between 1 and 100.
(3 marks)
Q9
Twin primes are pairs of primes having a difference of 2 (e.g. 3 and 5; 17 and 19). Find the other twin primes between 1 and 100.
(2 marks)
Q10
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.
(2 marks)
Q11
Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?
(2 marks)
Q12
How many three-digit prime numbers can you make using each of 2, 4 and 5 once?
(2 marks)
Q13
3 is prime and 2 × 3 + 1 = 7 is also prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples.
(3 marks)
4
EXERCISE 5.4 · 13 QUESTIONS
Co-prime Numbers and Prime Factorisation
Q1
Check if these pairs are safe (treasures on both numbers, jump size 1 not allowed): (a) 15 and 39 (b) 4 and 15 (c) 18 and 29 (d) 20 and 55.
(3 marks)
Q2
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
(2 marks)
Q3
Give examples of number pairs where the first common multiple (of the idli-vada game) is the same as the product of the numbers, and examples where it is less than the product. How is it related to the pair being co-prime?
(3 marks)
Q4
Thread art: make pictures for (a) 15 pegs, thread-gap 10, (b) 10 pegs, thread-gap 7, (c) 14 pegs, thread-gap 6, (d) 8 pegs, thread-gap 3. Is the thread tied to every peg, and how is it related to the numbers being co-prime?
(3 marks)
Q5
Find the prime factorisations of: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000.
(3 marks)
Q6
The prime factorisation of a number has one 2, two 3s and one 11. What is the number?
(2 marks)
Q7
Find three prime numbers, all less than 30, whose product is 1955.
(2 marks)
Q8
Find the prime factorisation of these numbers without multiplying first: (a) 56 × 25 (b) 108 × 75 (c) 1000 × 81
(3 marks)
Q9
What is the smallest number whose prime factorisation has (a) three different prime numbers? (b) four different prime numbers?
(2 marks)
Q10
Are the following pairs co-prime? Guess first, then use prime factorisation: (a) 30 and 45 (b) 57 and 85 (c) 121 and 1331 (d) 343 and 216
(3 marks)
Q11
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
(3 marks)
Q12
The first number has prime factorisation 2 × 3 × 7 and the second 3 × 7 × 11. Are they co-prime? Does one of them divide the other?
(3 marks)
Q13
Guna says, 'Any two prime numbers are co-prime.' Is he right?
(2 marks)
5
EXERCISE 5.5 · 12 QUESTIONS
Divisibility Rules
Q1
Is 8536 divisible by 4?
(2 marks)
Q2
Consider: (1) Only the last two digits matter when deciding if a 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?
(2 marks)
Q3
Find numbers between 120 and 140 that are divisible by 8. Also between 1120 and 1140, and 3120 and 3140. What do you observe?
(2 marks)
Q4
Change the last two digits of 8560 so that the resulting number is a multiple of 8.
(2 marks)
Q5
Do you agree? (1) Only the last three digits matter when deciding if a number is divisible by 8. (2) If the number formed by the last three digits is divisible by 8, the original number is divisible by 8. (3) If the original number is divisible by 8, the number formed by the last three digits is divisible by 8.
(2 marks)
Q6
2024 is a leap year. Leap years occur in years that are multiples of 4, except those divisible by 100 but not by 400. (a) From the year you were born till now, which years were leap years? (b) From 2024 till 2099, how many leap years are there?
(3 marks)
Q7
Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.
(3 marks)
Q8
Explore if each statement is always, sometimes or never true: (a) The sum of two even numbers gives a multiple of 4. (b) The sum of two odd numbers gives a multiple of 4.
(2 marks)
Q9
Find the remainders when each of the numbers 78, 99, 173, 572, 980, 1111, 2345 is divided by (i) 10, (ii) 5, (iii) 2.
(5 marks)
Q10
The teacher asked if 14560 is divisible by all of 2, 4, 5, 8 and 10. Guna checked only two of these numbers and declared it divisible by all. What could those two numbers be?
(2 marks)
Q11
Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160?
(3 marks)
Q12
Write two numbers whose product is 10000. The two numbers should not have 0 as their units digit.
(3 marks)