Class 6 › Mathematics › Ganita Prakash Prime Time Class 6 Mathematics · Ganita Prakash · Chapter 5
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View all 6 amazing Marks Booster kits → This chapter develops factors, multiples and prime numbers through games. The idli-vada game introduces common multiples of two numbers, and treasure-hunting jumps introduce common factors and divisors. A prime number has exactly two factors \u2014 1 and itself \u2014 so the first primes are \(2, 3, 5, 7, 11, 13, \dots\); numbers with more than two factors are composite, and 1 is neither prime nor composite. The Sieve of Eratosthenes is used to list all primes up to 100. Two numbers are co-prime if their only common factor is 1, for example 4 and 9. The chapter introduces prime factorisation: every number greater than 1 can be written as a product of primes in only one way apart from order, e.g. \(56 = 2 \times 2 \times 2 \times 7\). Prime factorisation is then used to test whether two numbers are co-prime and whether one number divides another. Finally, simple divisibility tests for 10, 5, 2, 4 and 8 are derived by looking at the last one, two or three digits. The chapter also explores perfect numbers, twin primes and prime puzzles.
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 \(2, 3, 5, 7, 11\).
Composite number
A number with more than two factors, such as \(4, 6, 8, 9, 10\).
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. \(84 = 2 \times 2 \times 3 \times 7\).
Q1 What is a prime number? Answer Q2 What is a composite number? Answer Q3 Is 1 a prime or a composite number? Answer Q4 What is the only even prime number? Answer Q5 What does it mean for two numbers to be co-prime? Answer Q6 What is the prime factorisation of 56? Answer Q7 What is the Sieve of Eratosthenes? Answer Q8 How can prime factorisation tell if two numbers are co-prime? Answer Q9 Numbers divisible by 5 end in which digits? Answer Q10 How do you test if a number is divisible by 4? Answer Q1 1. At what number is ‘idli-vada’ said for the 10th time? Answer Q2 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’)? Answer Q3 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’)? Answer Q4 2. If the game is played for the numbers 1 to 90, find out: c. How many times would the children say ‘idli-vada’? Answer Q5 3. What if the game was played till 900? How would your answers change? Answer Q6 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. Answer Q1 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 Answer Q2 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 Answer Q3 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 Answer Q1 Which of the following could be the other number: 2, 3, 5, 8, 10? Answer Q1 What jump size can reach both 15 and 30? There are multiple jump sizes possible. Try to find them all. Answer Q1 Look at the table below. What do you notice? 1. Is there anything common among the shaded numbers? Answer Q2 Look at the table below. What do you notice? 2. Is there anything common among the circled numbers? Answer Q3 Look at the table below. What do you notice? 3. Which numbers are both shaded and circled? What are these numbers called? Answer Q1 1. Find all multiples of 40 that lie between 310 and 410. Answer Q2 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. Answer Q3 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. Answer Q4 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. Answer Q5 4. Find the common factors of: a. 20 and 28 Answer Q6 4. Find the common factors of: b. 35 and 50 Answer Q7 4. Find the common factors of: c. 4, 8 and 12 Answer Q8 4. Find the common factors of: d. 5, 15 and 25 Answer Q9 5. Find any three numbers that are multiples of 25 but not multiples of 50. Answer Q10 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’? Answer Q11 7. In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers? Answer Q12 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 ____ Answer Q13 9. Find the smallest number that is a multiple of all the numbers from 1 to 10, except for 7. Answer Q14 10. Find the smallest number that is a multiple of all the numbers from 1 to 10. Answer Q1 1. We see that 2 is a prime and also an even number. Is there any other even prime? Answer Q2 2. Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference? Answer Q3 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? Answer Q4 4. Which of the following numbers are prime: 23, 51, 37, 26? Answer Q5 5. Write three pairs of prime numbers less than 20 whose sum is a multiple of 5. Answer Q6 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. Answer Q7 7. Find seven consecutive composite numbers between 1 and 100. Answer Q8 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. Answer Q9 9. Identify whether each statement is true or false. Explain. a. There is no prime number whose units digit is 4. Answer Q10 9. Identify whether each statement is true or false. Explain. b. A product of primes can also be prime. Answer Q11 9. Identify whether each statement is true or false. Explain. c. Prime numbers do not have any factors. Answer Q12 9. Identify whether each statement is true or false. Explain. d. All even numbers are composite numbers. Answer Q13 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. Answer Q14 10. Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330? Answer Q15 11. How many three-digit prime numbers can you make using each of 2, 4 and 5 once? Answer Q16 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. Answer Q1 Check if these pairs are safe: a. 15 and 39 Answer Q2 Check if these pairs are safe: b. 4 and 15 Answer Q3 Check if these pairs are safe: c. 18 and 29 Answer Q4 Check if these pairs are safe: d. 20 and 55 Answer Q1 Which of the following pairs of numbers are co-prime? a. 18 and 35 Answer Q2 Which of the following pairs of numbers are co-prime? b. 15 and 37 Answer Q3 Which of the following pairs of numbers are co-prime? c. 30 and 415 Answer Q4 Which of the following pairs of numbers are co-prime? d. 17 and 69 Answer Q5 Which of the following pairs of numbers are co-prime? e. 81 and 18 Answer Q1 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. Answer Q2 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. Answer Q1 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 64 Answer Q2 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 104 Answer Q3 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 105 Answer Q4 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 243 Answer Q5 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 320 Answer Q6 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 141 Answer Q7 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 1728 Answer Q8 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 729 Answer Q9 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 1024 Answer Q10 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 1331 Answer Q11 1. Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000. 1000 Answer Q12 2. The prime factorisation of a number has one 2, two 3s, and one 11. What is the number? Answer Q13 3. Find three prime numbers, all less than 30, whose product is 1955. Answer Q14 4. Find the prime factorisation of these numbers without multiplying first a. 56 × 25 Answer Q15 4. Find the prime factorisation of these numbers without multiplying first b. 108 × 75 Answer Q16 4. Find the prime factorisation of these numbers without multiplying first c. 1000 × 81 Answer Q17 5. What is the smallest number whose prime factorisation has: a. three different prime numbers? Answer Q18 5. What is the smallest number whose prime factorisation has: b. four different prime numbers? Answer Q1 1. Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. a. 30 and 45 Answer Q2 1. Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. b. 57 and 85 Answer Q3 1. Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. c. 121 and 1331 Answer Q4 1. Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. d. 343 and 216 Answer Q5 2. Is the first number divisible by the second? Use prime factorisation. a. 225 and 27 Answer Q6 2. Is the first number divisible by the second? Use prime factorisation. b. 96 and 24 Answer Q7 2. Is the first number divisible by the second? Use prime factorisation. c. 343 and 17 Answer Q8 2. Is the first number divisible by the second? Use prime factorisation. d. 999 and 99 Answer Q9 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? Answer Q10 4. Guna says, “Any two prime numbers are co-prime?”. Is he right? Answer Q1 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? Answer Q2 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? Answer Q3 2. Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes. Answer Q4 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. Answer Q5 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. Answer Q6 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 Answer Q7 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 Answer Q8 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 Answer Q9 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? Answer Q10 6. Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160. Answer Q11 7. Write two numbers whose product is 10000. The two numbers should not have 0 as the units digit. Answer
A prime number has exactly how many factors?
A Three B Two C One D Four
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