Prime Numbers — Questions and Answers
Welcome to HSLC Guru. This page gives complete, worked solutions to every Work it Out exercise of ASSEB Class 6 New Mathematics Chapter 5, Prime Number, along with extra practice questions in simple English.
Summary
This chapter begins with factors and multiples. If one number is a multiple of another, then the second number is called a factor of the first; for example, 6 is a multiple of 3, so 3 is a factor of 6. A factor shared by two or more numbers is a common factor, and a multiple shared by them is a common multiple. The “Friendship Game” and the “Jumping Competition” help us discover these ideas.
A number with exactly two factors (1 and itself) is a prime number, while a number with more than two factors is a composite number. The number 1 is neither prime nor composite. Using the Sieve of Eratosthenes we find that there are 25 prime numbers from 1 to 100. Two numbers with no common factor other than 1 are called co-prime numbers.
Every composite number can be written as a product of prime numbers — this is its prime factorisation, and it is unique apart from the order of the factors. The factor tree is a handy way to find it. The chapter also covers the divisibility tests for 2, 3, 4, 5, 6, 7, 8, 9 and 10, together with perfect numbers, twin primes, prime triplets, mirror numbers and the famous Ramanujan magic square.
Summary: This page gives complete, worked answers to every Work it Out exercise (5.1 to 5.8) and the Fun with Numbers activities of ASSEB Class 6 New Mathematics Chapter 5, Prime Number. It covers factors and multiples, common factors and common multiples, prime and composite numbers, co-prime numbers, the Sieve of Eratosthenes, prime factorisation using the factor tree, divisibility tests for 2, 3, 4, 5, 6, 7, 8, 9 and 10, perfect numbers, twin primes, prime triplets and the Ramanujan magic square, with extra MCQs, fill in the blanks, true or false and short answer questions.
Textbook Questions and Answers
Work it Out 5.1
In the Friendship Game the cards are numbered 1 to 40. Kunal says “Hi” on multiples of 2, Rahul says “Hello” on multiples of 5, and they shake hands when both speak together — that is, on the common multiples of 2 and 5, i.e. the multiples of 10.
1. At which card number do Kunal and Rahul shake hands for the fifth time?
Answer: A handshake happens on multiples of 10. From 1 to 40 the multiples of 10 are 10, 20, 30 and 40 — so only 4 handshakes are possible. A fifth handshake would occur at 50, but since this pack has no card larger than 40, a fifth handshake is not possible in this game.
2. How many times did Kunal say “Hi” only?
Answer: From 1 to 40 there are 20 multiples of 2 (2, 4, …, 40). Of these, 4 are multiples of 10 (10, 20, 30, 40) where a handshake happens. So “Hi” alone is said 20 − 4 = 16 times.
3. How many times did Rahul say “Hello” only?
Answer: From 1 to 40 there are 8 multiples of 5 (5, 10, 15, 20, 25, 30, 35, 40). Of these, 4 are multiples of 10. So “Hello” alone is said 8 − 4 = 4 times.
4. How many times did they shake hands?
Answer: On the multiples of 10 (10, 20, 30, 40) — a total of 4 handshakes.
5. Observe the multiples of 2 and 5 in the table. How can we relate this table with the “Friendship Game”? Extend the table up to 40.
Answer: Each row of the table lists the multiples of one number. The row of multiples of 2 gives Kunal’s “Hi” numbers and the row of multiples of 5 gives Rahul’s “Hello” numbers. The numbers appearing in both rows (the common multiples) are exactly where they shake hands. Extended to 40 — multiples of 2: 2, 4, 6, …, 40; multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40; common multiples: 10, 20, 30, 40.
6. Play the game with the pairs: (a) 3, 5 (b) 4, 7 (c) 2, 6 (d) 3, 6. At which number is the first handshake?
Answer: The first handshake occurs at the smallest common multiple —
- (a) 3 and 5: common multiples 15, 30, … → first at 15.
- (b) 4 and 7: common multiples 28, 56, … → first at 28.
- (c) 2 and 6: common multiples 6, 12, 18, … → first at 6.
- (d) 3 and 6: common multiples 6, 12, 18, … → first at 6.
7. From the diagram of multiples of 3 and multiples of 4, find the common multiples of 3 and 4.
Answer: Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, …; multiples of 4: 4, 8, 12, 16, 20, 24, …. The numbers common to both — the common multiples of 3 and 4 — are 12, 24, 36, … (that is, the multiples of 12).
Work it Out 5.2
1. Consider four strings of pearls A, B, C and D. A: 50, 45, 40, 35, …; B: 44, 42, 40, 38, …; C: 70, 63, 56, 49, …; D: 24, 27, 30, 33, 36, 39, 42, 45, 48. (i) Identify the special pattern on each string. (ii) Which number does not lie on the intersection of any strings? (a) 30 (b) 36 (c) 42 (d) 39. (iii) Is there a special pattern in the numbers lying in the intersection of two strings?
Answer: (i) A = multiples of 5, B = multiples of 2 (even numbers), C = multiples of 7 and D = multiples of 3.
(ii) String D (multiples of 3) crosses A, B and C. 30 is a common multiple of 3 and 5 (meets A), 36 and 42 are even (meet B), and 42 is also a multiple of 7 (meets C). But 39 is a multiple of 3 only — it is odd and not a multiple of 5 or 7 — so (d) 39 lies on no intersection.
(iii) Yes. Every number at an intersection is a common multiple of the base numbers of the two strings.
2. A number for which the sum of all its factors equals twice the number is a Perfect Number (e.g. 6: factors 1, 2, 3, 6; sum 1+2+3+6 = 12 = twice 6). Check whether 15 and 28 are perfect numbers.
Answer: For 15: factors 1, 3, 5, 15; sum = 1+3+5+15 = 24, but twice 15 = 30. Since 24 ≠ 30, 15 is not perfect. For 28: factors 1, 2, 4, 7, 14, 28; sum = 1+2+4+7+14+28 = 56 = twice 28. So 28 is a perfect number.
3. Find the common factors of: (a) 12, 18 (b) 6, 9, 15 (c) 24, 36 (d) 13, 17
- (a) 12, 18 → common factors 1, 2, 3, 6.
- (b) 6, 9, 15 → common factors 1, 3.
- (c) 24, 36 → common factors 1, 2, 3, 4, 6, 12.
- (d) 13, 17 → both prime, common factor only 1.
4. A and B are two numbers, both less than 30. 7 is their common factor. If A + B = 49, find A and B.
Answer: The multiples of 7 less than 30 are 7, 14, 21, 28. The pair adding to 49 is 21 + 28 = 49. So A = 21 and B = 28 (or the other way round).
5. Find all the factors of 150 which are less than 30.
Answer: The factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150. Those less than 30 are 1, 2, 3, 5, 6, 10, 15, 25.
6. Find all the multiples of 30 that lie between 200 and 300.
Answer: 30 × 7 = 210, 30 × 8 = 240, 30 × 9 = 270. So the multiples of 30 between 200 and 300 are 210, 240, 270.
7. The teacher sets two numbers for Papiya and Sukanya, both less than 10, so that their first handshake occurs only after card 50 (the pack has 100 cards). What are the possible pairs?
Answer: Both numbers must be less than 10 and their smallest common multiple (LCM) must be greater than 50. Such pairs are (7, 8) with LCM 56, (7, 9) with LCM 63, and (8, 9) with LCM 72.
8. In the jumping game Labhita places dolls on 56 and 80. What jump sizes are needed to land on both numbers?
Answer: The jump size must be a common factor of 56 and 80. Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56; factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80. The common factors are 1, 2, 4, 8 — these are the possible jump sizes.
9. Raju’s age is a common multiple of 5 and 3 and is a two-digit number. One of the digits is 2 more than the other. Find his age.
Answer: A common multiple of 5 and 3 is a multiple of 15. The two-digit multiples of 15 are 15, 30, 45, 60, 75, 90. The one whose digits differ by 2 is 75 (7 = 5 + 2). So Raju’s age is 75 years.
10. Match Column I with Column II: (i) 35 (ii) 15 (iii) 16 (iv) 20; (a) A multiple of 8 (b) A factor of 40 (c) A common multiple of 3 and 5 (d) A factor of 70.
| Column I | Column II |
|---|---|
| (i) 35 | (d) A factor of 70 |
| (ii) 15 | (c) A common multiple of 3 and 5 |
| (iii) 16 | (a) A multiple of 8 |
| (iv) 20 | (b) A factor of 40 |
11. Find only three numbers that are multiples of 10 but not multiples of 15.
Answer: Multiples of 10 are 10, 20, 30, 40, …. Three of these that are not multiples of 15 are 10, 20, 40 (30 and 60 are multiples of 15, so they are excluded).
12. Find the smallest number that is a multiple of all the numbers from 1 to 8 except 7.
Answer: We need the LCM of 1, 2, 3, 4, 5, 6, 8. Using prime factors, LCM = 2³ × 3 × 5 = 8 × 15 = 120, which is divisible by each of 1, 2, 3, 4, 5, 6 and 8.
13. Find the smallest number that is a multiple of all the numbers from 1 to 10.
Answer: The LCM of 1 to 10 = 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2520.
14. A machine has inlets A and B and outlet C. When two numbers are put in A and B, their common multiples come out of C. If the outputs are 18, 36 and 42, what are the possible inputs?
Answer: Since 18, 36 and 42 are all common multiples of A and B, the LCM of A and B must divide all three. The greatest common factor of 18, 36 and 42 is 6, so the LCM of A and B is 6. A suitable pair is 2 and 3 (LCM 6, whose common multiples 6, 12, 18, 24, 30, 36, 42, … include 18, 36 and 42). Pairs such as 1 and 6, or 6 and 2, also work.
Work it Out 5.3
1. How many prime and composite numbers are there from 1 to 40?
Answer: The primes from 1 to 40 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 — that is 12 primes. The number 1 is neither prime nor composite, so the composite numbers = 40 − 1 − 12 = 27. Thus there are 12 primes and 27 composite numbers.
2. Section A shows 5 multiples of 6, section B shows multiples of a particular number, and 42 lies in the common region. Identify the particular number and write four other multiples of it.
Answer: The common number 42 = 6 × 7, so 42 is a common multiple of 6 and 7. Hence the particular number is 7. Four other multiples of 7 are 14, 21, 28, 35.
Work it Out 5.4 (Sieve of Eratosthenes)
Using the Sieve of Eratosthenes, the 25 prime numbers from 1 to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
1. Which is the only even number among the circled (prime) numbers?
Answer: The only even prime number is 2.
2. What are the smallest and largest differences between two consecutive primes in the table?
Answer: The smallest difference is 1 (between 2 and 3); the largest difference is 8 (between 89 and 97).
3. Does each row of the table contain an equal number of primes?
Answer: No. The number of primes per row is — 1–10: 4, 11–20: 4, 21–30: 2, 31–40: 2, 41–50: 3, 51–60: 2, 61–70: 2, 71–80: 3, 81–90: 2, 91–100: 1. So the rows do not contain an equal number of primes.
4. Identify the rows containing the fewest and the most prime numbers.
Answer: The rows 1–10 and 11–20 have the most primes (4 each), while the row 91–100 has the fewest (only 1).
5. Which of the following are prime? 11, 51, 73, 91 and 99.
Answer: 11 and 73 are prime. 51 = 3 × 17, 91 = 7 × 13 and 99 = 9 × 11 are all composite.
6. Express as a sum of three prime numbers: 10, 26, 63.
Answer: 10 = 2 + 3 + 5; 26 = 2 + 11 + 13; 63 = 3 + 29 + 31.
7. 13 and 31 are primes with the same digits 1 and 3. Find such pairs of primes up to 100.
Answer: Such pairs are 13 and 31, 17 and 71, 37 and 73, 79 and 97.
8. Identify five consecutive composite numbers between 1 and 100.
Answer: 24, 25, 26, 27, 28 are five consecutive composite numbers (the next prime is 29). Another example is 90, 91, 92, 93, 94, 95, 96.
9. Find pairs of primes in the table whose difference is 2 (twin primes).
Answer: The twin-prime pairs are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61) and (71, 73).
10. State whether each statement is true or false: (a) 2 is the only even prime number. (b) The sum of two prime numbers is also prime. (c) The product of two prime numbers is also prime. (d) A prime number has more than two factors. (e) There are infinitely many primes. (f) All odd numbers are composite numbers.
- (a) True.
- (b) False (e.g. 3 + 5 = 8, which is composite).
- (c) False (e.g. 2 × 3 = 6, which is composite).
- (d) False (a prime has exactly two factors).
- (e) True.
- (f) False (e.g. 9, 15, 21 are odd but composite).
11. Express each as the product of three distinct prime numbers: 30, 42, 70 and 105.
Answer: 30 = 2 × 3 × 5; 42 = 2 × 3 × 7; 70 = 2 × 5 × 7; 105 = 3 × 5 × 7.
12. How many three-digit prime numbers can be formed using each of the digits 2, 3 and 5 exactly once?
Answer: The possible numbers are 235, 253, 325, 352, 523, 532. Of these only 523 is prime. So exactly 1 three-digit prime number can be formed.
13. 2 is prime and 2 × 2 + 1 = 5 is also prime. Find other primes for which doubling and adding 1 gives another prime. Find at least three.
Answer: 3 (2 × 3 + 1 = 7), 5 (2 × 5 + 1 = 11) and 11 (2 × 11 + 1 = 23) are three such primes. Others are 23 (→ 47) and 29 (→ 59).
Work it Out 5.5 (Art of Co-Prime Numbers)
When pegs are placed at equal gaps around a circle and a thread is tied taking a fixed gap, the thread can touch all the pegs only if the number of pegs and the gap are co-prime.
1. Draw a similar diagram for: (i) number of pegs = 16, thread gap = 4; (ii) number of pegs = 16, thread gap = 7.
Answer: (i) 16 and 4 are not co-prime (their greatest common factor is 4). So the thread does not touch all the pegs — starting from peg 1 with a gap of 4 it touches only 1, 5, 9, 13 and returns to the start, covering just 4 of the 16 pegs.
(ii) 16 and 7 are co-prime (greatest common factor 1). So the thread touches all 16 pegs, forming a 16-pointed star: 1 → 8 → 15 → 6 → 13 → 4 → 11 → 2 → 9 → 16 → 7 → 14 → 5 → 12 → 3 → 10 → 1.
Work it Out 5.6 (Prime Factorisation)
1. Find the prime factorisation of: 32, 63, 89, 108, 250, 540, 729, 915, 1728 and 1331.
- 32 = 2 × 2 × 2 × 2 × 2 = 2⁵
- 63 = 3 × 3 × 7 = 3² × 7
- 89 = 89 (it is a prime number)
- 108 = 2 × 2 × 3 × 3 × 3 = 2² × 3³
- 250 = 2 × 5 × 5 × 5 = 2 × 5³
- 540 = 2 × 2 × 3 × 3 × 3 × 5 = 2² × 3³ × 5
- 729 = 3 × 3 × 3 × 3 × 3 × 3 = 3⁶
- 915 = 3 × 5 × 61
- 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2⁶ × 3³
- 1331 = 11 × 11 × 11 = 11³
2. The prime factorisation of a number has two 2’s, three 3’s and one 5. What is the number?
Answer: The number = 2 × 2 × 3 × 3 × 3 × 5 = 2² × 3³ × 5 = 540.
3. What is the smallest number whose prime factorisation has (a) three different primes? (b) four different primes?
Answer: (a) Using the smallest three primes 2, 3, 5 — the number is 2 × 3 × 5 = 30. (b) Using 2, 3, 5, 7 — the number is 2 × 3 × 5 × 7 = 210.
4. Find the prime factorisation without multiplying first: (a) 55 × 56 (b) 36 × 97 (c) 1000 × 144.
- (a) 55 × 56 = (5 × 11) × (2³ × 7) = 2³ × 5 × 7 × 11
- (b) 36 × 97 = (2² × 3²) × 97 = 2² × 3² × 97
- (c) 1000 × 144 = (2³ × 5³) × (2⁴ × 3²) = 2⁷ × 3² × 5³
Work it Out 5.7 (Divisibility by Prime Factorisation)
1. Is the first number divisible by the second (use prime factorisation)? (a) 225 and 45 (b) 999 and 99 (c) 1024 and 32 (d) 1331 and 33.
- (a) 225 = 3² × 5², 45 = 3² × 5. The factorisation of 45 is fully contained in 225 → divisible (225 ÷ 45 = 5).
- (b) 999 = 3³ × 37, 99 = 3² × 11. Since 11 is not a factor of 999 → not divisible.
- (c) 1024 = 2¹⁰, 32 = 2⁵. Fully contained → divisible (1024 ÷ 32 = 32).
- (d) 1331 = 11³, 33 = 3 × 11. Since 3 is not a factor of 1331 → not divisible.
2. Chinmoy says, “Any two prime numbers are co-prime.” Is he correct?
Answer: Yes, he is correct. The only factors of a prime are 1 and itself, so two different primes share no common factor other than 1. Hence any two distinct primes are always co-prime.
3. Are the following pairs co-prime? Can you spot a pattern? (a) 1 × 2 + 41, 2 × 3 + 41 (b) 2 × 3 + 41, 3 × 4 + 41 (c) 3 × 4 + 41, 4 × 5 + 41 (d) 28 × 29 + 41, 29 × 30 + 41. Also, are 34 × 35 + 41 and 35 × 36 + 41 co-prime?
- (a) 43 and 47 — both prime → co-prime.
- (b) 47 and 53 — both prime → co-prime.
- (c) 53 and 61 — both prime → co-prime.
- (d) 853 and 911 — both prime → co-prime.
Answer: Each number has the form n × (n+1) + 41 = n² + n + 41 (Euler’s famous prime-generating polynomial), which gives a prime for every n from 1 to 39, and consecutive terms are always co-prime. So 34 × 35 + 41 = 1231 and 35 × 36 + 41 = 1301 are both prime and hence co-prime.
4. Are 5 × 6 × 7 × 10 and 14 × 25 co-prime? Is one divisible by the other?
Answer: 5 × 6 × 7 × 10 = 2100 = 2² × 3 × 5² × 7 and 14 × 25 = 350 = 2 × 5² × 7. They share the prime factors 2, 5 and 7, so they are not co-prime. Since the factorisation of 350 is fully contained in 2100, 2100 is divisible by 350 (2100 = 6 × 350).
Work it Out 5.8 (Test of Divisibility)
1. What are the largest and smallest 4-digit numbers divisible by 4?
Answer: The smallest 4-digit number 1000 is divisible by 4 (1000 ÷ 4 = 250), so the smallest is 1000. For the largest, 4 × 2499 = 9996, so the largest is 9996.
2. Decide whether each statement is always true, sometimes true or never true: (a) The sum of two odd numbers is a multiple of 2. (b) The sum of two even numbers is a multiple of 5. (c) The sum of two odd numbers is a multiple of 6.
- (a) Always true — odd + odd = even, which is always a multiple of 2 (e.g. 3 + 5 = 8).
- (b) Sometimes true — e.g. 2 + 8 = 10 (a multiple of 5), but 2 + 4 = 6 (not a multiple of 5).
- (c) Sometimes true — e.g. 3 + 3 = 6 (a multiple of 6), but 3 + 5 = 8 (not a multiple of 6).
3. If a number is divisible by 10, will it be divisible by both 5 and 2? Justify.
Answer: Yes. Since 10 = 2 × 5, and both 2 and 5 are factors of 10, they are factors of every multiple of 10 too. Also, a number divisible by 10 ends in 0, and any number ending in 0 is divisible by 2 (even) and by 5.
4. Rajesh must check whether 23480 is divisible by all of 2, 4, 5, 8 and 10, but is allowed to test only two of these. Which two should he choose?
Answer: 8 and 10. A number divisible by 8 is automatically divisible by 2 and 4 (both are factors of 8); a number divisible by 10 is automatically divisible by 2 and 5. So testing 8 and 10 covers 2, 4, 5, 8 and 10. (For 23480: the last three digits 480 are divisible by 8, and it ends in 0, so it is indeed divisible by all of them.)
5. Which of the following are divisible by all of 2, 4, 5, 8 and 10? 3450, 640, 9000, 293442, 1645160.
Answer: Being divisible by all of them means being divisible by their LCM, 40 (that is, ending in 0 and having the last three digits divisible by 8). 3450 fails (450 is not divisible by 8) and 293442 fails (it does not end in 0). The rest — 640, 9000 and 1645160 — are divisible by all of them.
6. Find the remainders when each number is divided by (i) 10, (ii) 5 and (iii) 2: 89, 174, 299, 590, 1254, 3451.
| Number | Remainder ÷ 10 | Remainder ÷ 5 | Remainder ÷ 2 |
|---|---|---|---|
| 89 | 9 | 4 | 1 |
| 174 | 4 | 4 | 0 |
| 299 | 9 | 4 | 1 |
| 590 | 0 | 0 | 0 |
| 1254 | 4 | 4 | 0 |
| 3451 | 1 | 1 | 1 |
7. A leap year is a multiple of 4; a centurial year (100, 200, 1000, 2000, …) must also be divisible by 400 to be a leap year. (a) Write your birth year and check whether it is a leap year. (b) Which years were leap years from your birth year till now? (c) How many leap years are there from 2004 to 2084?
Answer: (a) Divide the birth year by 4; if it divides exactly (and, for a centurial year, is also divisible by 400) it is a leap year. Example: 2012 ÷ 4 = 503 and it is not a centurial year, so 2012 is a leap year. (b) Every multiple of 4 after your birth year (with centurial years also needing 400) is a leap year. (c) The multiples of 4 from 2004 to 2084 are 2004, 2008, …, 2084, giving (2084 − 2004) ÷ 4 + 1 = 21 years. There is no centurial year in this range, so all 21 are leap years.
Fun with Numbers
1. How is the factor-spotting game played with 30 cards numbered 1 to 30?
Answer: It is a game for two players A and B. One player picks a card (say 28) and keeps it; the other then collects all the remaining cards whose numbers are factors of it (1, 2, 4, 7, 14). Players take turns like this until all cards are used up. Each player adds the numbers on the cards they collected, and the one with the greater sum wins. A good strategy is to pick numbers with few factors (or primes) so the opponent gets small numbers.
2. The box contains four numbers — first row: 3, 5; second row: 105, 7. Which statements are correct? (i) The numbers in the first row are co-prime. (ii) The numbers in the first column are co-prime. (iii) The numbers in the second row are not co-prime. (iv) The numbers in the second column are not co-prime. (v) All the prime factors of 105 are present in the box.
- (i) 3 and 5 are co-prime → correct.
- (ii) 3 and 105: since 105 = 3 × 5 × 7, their common factor is 3 — not co-prime, so the statement is incorrect.
- (iii) 105 and 7: common factor 7 — not co-prime, so the statement is correct.
- (iv) 5 and 7 are both prime and so co-prime — the statement “not co-prime” is incorrect.
- (v) 105 = 3 × 5 × 7; the primes 3, 5 and 7 are all in the box → correct.
So the correct statements are (i), (iii) and (v).
3. Look at Srinivasa Ramanujan’s magic square: 22, 12, 18, 87 / 88, 17, 9, 25 / 10, 24, 89, 16 / 19, 86, 23, 11. (a) What is the sum of each row, column and diagonal? (b) Is it the same every time? (c) Find the prime and composite numbers. (d) Identify co-prime numbers in each row. (e) Why is the first row 22-12-1887 special?
- (a) Every row, column and diagonal adds up to 139 (e.g. 22 + 12 + 18 + 87 = 139).
- (b) Yes, the sum is the same (139) every time — that is why it is a magic square.
- (c) Prime numbers: 11, 17, 19, 23, 89. The rest (9, 10, 12, 16, 18, 22, 24, 25, 86, 87, 88) are composite.
- (d) Examples of co-prime pairs in each row — Row 1: 22 and 87; Row 2: 17 and 25; Row 3: 89 and 16; Row 4: 19 and 86 (each such pair has 1 as its only common factor).
- (e) 22-12-1887 is the birth date of Srinivasa Ramanujan (22 December 1887).
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. The smallest prime number is — (a) 1 (b) 2 (c) 3 (d) 0
Answer: (b) 2
2. Which of the following is a composite number? (a) 23 (b) 29 (c) 21 (d) 31
Answer: (c) 21
3. The number 1 is — (a) prime (b) composite (c) neither prime nor composite (d) a twin prime
Answer: (c) neither prime nor composite
4. How many prime numbers are there from 1 to 100? (a) 24 (b) 25 (c) 26 (d) 20
Answer: (b) 25
5. The prime factorisation of 36 is — (a) 2 × 2 × 3 × 3 (b) 2 × 3 × 3 × 3 (c) 2 × 2 × 2 × 3 (d) 6 × 6
Answer: (a) 2 × 2 × 3 × 3
6. Which pair is co-prime? (a) 12 and 18 (b) 8 and 12 (c) 9 and 16 (d) 15 and 20
Answer: (c) 9 and 16
7. Which of the following is divisible by 4? (a) 122 (b) 234 (c) 316 (d) 150
Answer: (c) 316
8. A pair of twin primes is — (a) 3, 4 (b) 7, 11 (c) 11, 13 (d) 2, 3
Answer: (c) 11, 13
9. A number is divisible by 9 if the sum of its digits is divisible by — (a) 2 (b) 3 (c) 6 (d) 9
Answer: (d) 9
10. The smallest perfect number is — (a) 1 (b) 6 (c) 28 (d) 12
Answer: (b) 6
Fill in the Blanks
- A number with exactly two factors is called a ______ number. (prime)
- The number 1 is neither prime nor ______. (composite)
- The method used to find prime numbers is the ______ of Eratosthenes. (Sieve)
- A number ending in 0 or 5 is divisible by ______. (5)
- The only common factor of two co-prime numbers is ______. (1)
True or False
- 2 is the only even prime number. — True
- Every composite number can be written as a product of prime factors. — True
- 1 is the smallest prime number. — False (1 is not prime)
- Two prime numbers are always co-prime. — True
- All odd numbers are prime numbers. — False (9, 15 are odd but composite)
Short Answer Questions
1. What is the difference between a prime and a composite number?
Answer: A prime number has exactly two factors, 1 and itself (e.g. 2, 3, 5, 7), whereas a composite number has more than two factors (e.g. 4, 6, 8, 9).
2. Find the prime factorisation of 60.
Answer: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
3. Write the common factors of 24 and 36.
Answer: The common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12.
4. State the divisibility rule for 3 and check whether 156 is divisible by 3.
Answer: A number is divisible by 3 if the sum of its digits is divisible by 3. For 156, the digit sum is 1 + 5 + 6 = 12, which is divisible by 3 — so 156 is divisible by 3.
Key Terms
| Term | Meaning |
|---|---|
| Factor | A number that divides another number exactly (with no remainder) |
| Multiple | A number obtained by multiplying a given number by another |
| Common factor | A factor shared by two or more numbers |
| Common multiple | A multiple shared by two or more numbers |
| Prime number | A number with exactly two factors, 1 and itself |
| Composite number | A number with more than two factors |
| Co-prime numbers | Two numbers whose only common factor is 1 |
| Prime factorisation | Expressing a number as a product of its prime factors |
| Factor tree | A diagram that breaks a number down into its prime factors |
| Sieve of Eratosthenes | A method for finding all prime numbers up to a given number |
| Twin primes | Two primes with exactly one composite number between them |
| Perfect number | A number whose factor sum equals twice the number (e.g. 6, 28) |