Class 6 : Maths ( English ) – Lesson 5. Prime Time

EXPLANATION AND ANALYSIS

🌿 1. Introduction: Why “Prime Time” Matters

Numbers are not all the same. Some numbers can be broken into smaller equal parts easily, while some cannot. Understanding which numbers divide others and which do not is the main idea of this chapter. The lesson Prime Time helps students explore numbers deeply by studying factors, multiples, prime numbers, composite numbers, and divisibility rules.

🔵 This chapter sharpens logical thinking
🟢 It helps in simplifying calculations
🟡 It forms the base for fractions, LCM, and HCF
🔴 It prepares students for higher mathematics

🧠 2. Factors of a Number

A factor of a number is a number that divides it exactly, leaving no remainder.

🔹 If a number a divides b exactly, then a is a factor of b
🔹 Factors are always whole numbers
🔹 Every number has at least two factors: 1 and itself

🔵 Example: Factors of 12 are 1, 2, 3, 4, 6, 12
🟢 Here, 12 ÷ 3 = 4 (no remainder)

💡 Concept:
Factor × Factor = Number

✏️ Note:
1 is a factor of every number.

🌱 3. Finding Factors

Factors can be found by checking divisibility.

🔵 Start dividing the number by 1, 2, 3, …
🟢 Stop once the quotient becomes smaller than the divisor
🟡 Factors always come in pairs

🔹 Example:
For 18 → (1,18), (2,9), (3,6)

💡 Concept:
Factors occur in pairs.

🧠 4. Multiples of a Number

A multiple of a number is obtained by multiplying it by a whole number.

🔵 Multiples of 4 are 4, 8, 12, 16, …
🟢 Multiples are infinite
🟡 A number has unlimited multiples but limited factors

✏️ Note:
Every number is a multiple of itself.

🌿 5. Difference Between Factors and Multiples

Understanding the difference is very important.

🔵 Factors divide the number exactly
🟢 Multiples are obtained by multiplication
🟡 Number of factors is limited
🔴 Number of multiples is infinite

💡 Concept:
Factors divide, multiples multiply.

🧠 6. Prime Numbers

A prime number is a number greater than 1 that has exactly two factors.

🔹 The two factors are 1 and the number itself
🔹 Prime numbers cannot be divided by any other number

🔵 Examples: 2, 3, 5, 7, 11
🟢 2 is the only even prime number

💡 Concept:
Prime number → only two factors

✏️ Note:
1 is not a prime number because it has only one factor.

🌱 7. Composite Numbers

A composite number is a number that has more than two factors.

🔹 Composite numbers can be broken into smaller factors
🔹 They are not prime

🔵 Examples: 4, 6, 8, 9, 10
🟢 6 has factors 1, 2, 3, 6

💡 Concept:
Composite number → more than two factors

🧠 8. Identifying Prime and Composite Numbers

To identify whether a number is prime or composite:

🔵 List its factors
🟢 Count the number of factors
🟡 Two factors → prime
🔴 More than two factors → composite

✏️ Note:
2 and 3 are the smallest prime numbers.

🌿 9. Co-prime Numbers

Two numbers are called co-prime if they have no common factor other than 1.

🔵 Example: 8 and 15
🟢 Factors of 8: 1, 2, 4, 8
🟡 Factors of 15: 1, 3, 5, 15
🔴 Common factor = 1 only

💡 Concept:
Co-prime numbers may or may not be prime themselves.

🧠 10. Divisibility Rules

Divisibility rules help us check division quickly without full calculation.

🔵 A number is divisible by 2 if its last digit is even
🟢 A number is divisible by 3 if the sum of its digits is divisible by 3
🟡 A number is divisible by 5 if its last digit is 0 or 5
🔴 A number is divisible by 10 if its last digit is 0

✏️ Note:
Divisibility rules save time and reduce calculation errors.

🌍 11. Prime Numbers in Daily Life

Prime numbers play a role beyond school mathematics.

🔵 Used in computer security and coding
🟢 Used in number puzzles and games
🟡 Important in advanced mathematics

💡 Concept:
Prime numbers are the building blocks of all numbers.

🧠 12. Importance of Prime Time

This chapter helps students:

🔹 Understand number structure
🔹 Classify numbers logically
🔹 Prepare for LCM, HCF, and fractions
🔹 Develop reasoning skills

💡 Concept:
Strong basics of primes and factors make mathematics easier.

📘 Summary

The chapter Prime Time introduces the idea that numbers can be studied based on how they divide and multiply. Factors are numbers that divide a given number exactly, while multiples are obtained by multiplying the number. Prime numbers have exactly two factors, whereas composite numbers have more than two. The chapter also explains co-prime numbers and simple divisibility rules that make calculations faster.

By understanding factors, multiples, and primes, students learn how numbers are built. These ideas are essential for learning higher topics like fractions, LCM, and HCF. Prime Time builds strong number sense and logical thinking.

📝 Quick Recap

🟢 Factors divide a number exactly
🟡 Multiples are obtained by multiplication
🔵 Prime numbers have exactly two factors
🔴 Composite numbers have more than two factors
⚡ Co-prime numbers share only factor 1
🧠 Divisibility rules help in quick checking.

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TEXTBOOK QUESTIONS

🌿 FIGURE IT OUT

🌿 BASED ON MULTIPLES AND COMMON FACTORS

🔒 ❓ Question 1.
Find all multiples of 40 that lie between 310 and 410.

📌 ✅ Answer:
🔵 Step 1: 40 × 7 = 280 (less than 310, ignore)
🔵 Step 2: 40 × 8 = 320
🔵 Step 3: Add 40 successively → 360, 400
🔴 Next multiple 440 is greater than 410
✔️ Final: 320, 360, 400

🔒 ❓ Question 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:
🔵 Step 1: Multiples of 7 less than 40 → 7, 14, 21, 28, 35
🔵 Step 2: Find digit sums
🔹 7 → 7
🔹 14 → 1 + 4 = 5
🔹 21 → 2 + 1 = 3
🔹 28 → 2 + 8 = 10
🔹 35 → 3 + 5 = 8
✔️ Final: 35

🔒 ❓ (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:
🔵 Step 1: If 3 and 5 are factors, number must be a multiple of 15
🔵 Step 2: Multiples of 15 below 100 → 15, 30, 45, 60, 75, 90
🔵 Step 3: Check digits
🔹 45 → digits 4 and 5 (difference = 1)
✔️ Final: 45

🔒 ❓ Question 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:
🔵 Step 1: Check number 6
🔵 Step 2: Factors of 6 → 1, 2, 3, 6
🔵 Step 3: Sum = 1 + 2 + 3 + 6 = 12
🟡 Check: 12 = 2 × 6
✔️ Final: 6

🔒 ❓ Question 4. Find the common factors of:

🔒 ❓ (a) 20 and 28

📌 ✅ Answer:
🔵 Factors of 20 → 1, 2, 4, 5, 10, 20
🔵 Factors of 28 → 1, 2, 4, 7, 14, 28
✔️ Final: 1, 2, 4

🔒 ❓ (b) 35 and 50

📌 ✅ Answer:
🔵 Factors of 35 → 1, 5, 7, 35
🔵 Factors of 50 → 1, 2, 5, 10, 25, 50
✔️ Final: 1, 5

🔒 ❓ (c) 4, 8 and 12

📌 ✅ Answer:
🔵 Factors of 4 → 1, 2, 4
🔵 Factors of 8 → 1, 2, 4, 8
🔵 Factors of 12 → 1, 2, 3, 4, 6, 12
✔️ Final: 1, 2, 4

🔒 ❓ (d) 5, 15 and 25

📌 ✅ Answer:
🔵 Factors of 5 → 1, 5
🔵 Factors of 15 → 1, 3, 5, 15
🔵 Factors of 25 → 1, 5, 25
✔️ Final: 1, 5

🔒 ❓ Question 5.
Find any three numbers that are multiples of 25 but not multiples of 50.

📌 ✅ Answer:
🔵 Multiples of 25 → 25, 50, 75, 100, 125
🔴 Remove multiples of 50 → 50, 100
✔️ Final: 25, 75, 125

🔒 ❓ Question 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:
🔵 Step 1: ‘Idli-vada’ is spoken at common multiples
🔵 Step 2: First common multiple = LCM
🔵 Step 3: LCM must be greater than 50
🔵 Step 4: For numbers 8 and 9, LCM = 72
✔️ Final: 8 and 9

🔒 ❓ Question 7.
In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?

📌 ✅ Answer:
🔵 Factors of 28 → 1, 2, 4, 7, 14, 28
🔵 Factors of 70 → 1, 2, 5, 7, 10, 14, 35, 70
✔️ Final: 1, 2, 7, 14

🔒 ❓ Question 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.

📌 ✅ Answer:
🔵 Common multiples shown → 24, 48, 72
🔵 These numbers are multiples of 24
🔵 Possible original numbers → 6 and 8
✔️ Final: Multiples of 6 and multiples of 8

🔒 ❓ Question 9.
Find the smallest number that is a multiple of all the numbers from 1 to 10, except for 7.

📌 ✅ Answer:
🔵 Numbers considered → 1, 2, 3, 4, 5, 6, 8, 9, 10
🔵 LCM = 360
✔️ Final: 360

🔒 ❓ Question 10.
Find the smallest number that is a multiple of all the numbers from 1 to 10.

📌 ✅ Answer:
🔵 LCM of numbers from 1 to 10 = 2520
✔️ Final: 2520

🌿 BASED ON PRIME NUMBERS

🔒 ❓ Question 1.
We see that 2 is a prime and also an even number. Is there any other even prime?

📌 ✅ Answer:
🔵 Step 1: Any even number greater than 2 is divisible by 2
🔵 Step 2: A prime number has exactly two factors, 1 and itself
🔴 Step 3: Any even number greater than 2 has more than two factors
✔️ Final: No, 2 is the only even prime number

🔒 ❓ Question 2.
Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference?

📌 ✅ Answer:
🔵 Step 1: List primes till 100
🔵 Step 2: Find differences between consecutive primes
🔵 Step 3: Smallest difference → between 2 and 3 = 1
🔵 Step 4: Largest difference → between 89 and 97 = 8
✔️ Final: Smallest difference = 1, Largest difference = 8

🔒 ❓ Question 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:
🔵 Step 1: Observe the table of primes arranged in decades
🔵 Step 2: Count primes in each row
🔴 Step 3: Numbers of primes are not equal in every row
🔵 Step 4: Least primes → 1–10 and 90–100
🔵 Step 5: Most primes → 11–20 and 71–80
✔️ Final: No, primes are not equally distributed; least in 1–10 and 90–100, most in 11–20 and 71–80

🔒 ❓ Question 4.
Which of the following numbers are prime: 23, 51, 37, 26?

📌 ✅ Answer:
🔵 Step 1: 23 → factors: 1, 23
🔵 Step 2: 51 → divisible by 3 (5 + 1 = 6)
🔵 Step 3: 37 → factors: 1, 37
🔵 Step 4: 26 → divisible by 2
✔️ Final: 23 and 37 are prime

🔒 ❓ Question 5.
Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.

📌 ✅ Answer:
🔵 Step 1: List primes less than 20 → 2, 3, 5, 7, 11, 13, 17, 19
🔵 Step 2: Check sums
🔵 2 + 3 = 5
🔵 7 + 13 = 20
🔵 11 + 19 = 30
✔️ Final: (2, 3), (7, 13), (11, 19)

🔒 ❓ Question 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:
🔵 Step 1: Look for primes with reversed digits
🔵 Step 2: Check both numbers are prime
✔️ Final: (13, 31), (17, 71), (37, 73), (79, 97)

🔒 ❓ Question 7.
Find seven consecutive composite numbers between 1 and 100.

📌 ✅ Answer:
🔵 Step 1: Look for a continuous block with no primes
🔵 Step 2: Numbers 90 to 96 are all composite
✔️ Final: 90, 91, 92, 93, 94, 95, 96

🔒 ❓ Question 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:
🔵 Step 1: Check prime pairs with difference 2
✔️ Final: (5, 7), (11, 13), (29, 31), (41, 43), (59, 61), (71, 73)

🔒 ❓ Question 9.
Identify whether each statement is true or false. Explain.

🔒 ❓ (a) There is no prime number whose units digit is 4.

📌 ✅ Answer:
🔵 Any number ending in 4 is even and greater than 2
✔️ Final: True

🔒 ❓ (b) A product of primes can also be prime.

📌 ✅ Answer:
🔵 Product of two primes has more than two factors
✔️ Final: False

🔒 ❓ (c) Prime numbers do not have any factors.

📌 ✅ Answer:
🔵 Prime numbers have exactly two factors
✔️ Final: False

🔒 ❓ (d) All even numbers are composite numbers.

📌 ✅ Answer:
🔵 2 is even and prime
✔️ Final: False

🔒 ❓ (e) 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.

📌 ✅ Answer:
🔵 After any odd prime, the next number is even and greater than 2
✔️ Final: True

🔒 ❓ Question 10.
Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?

📌 ✅ Answer:
🔵 45 = 3 × 3 × 5 (not distinct)
🔵 60 = 2 × 2 × 3 × 5 (more than three primes)
🔵 91 = 7 × 13 (only two primes)
🔵 105 = 3 × 5 × 7
🔵 330 = 2 × 3 × 5 × 11 (four primes)
✔️ Final: 105

🔒 ❓ Question 11.
How many three-digit prime numbers can you make using each of 2, 4 and 5 once?

📌 ✅ Answer:
🔵 Step 1: Possible numbers → 245, 254, 425, 452, 524, 542
🔵 Step 2: Check divisibility
🔴 Numbers ending with 2, 4 are even
🔴 Numbers ending with 5 are divisible by 5
✔️ Final: 0

🔒 ❓ Question 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:
🔵 Step 1: Take a prime p
🔵 Step 2: Compute 2p + 1
✔️ Final:
2 → 5
3 → 7
5 → 11
11 → 23
23 → 47

🌿 BASED ON PRIME FACTORISATION

🔒 ❓ Question 1.
Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000.

📌 ✅ Answer:

🔵 64
🔵 Step 1: 64 is even → divide by 2
🔵 Step 2: 64 = 2 × 32 = 2 × 2 × 16 = 2 × 2 × 2 × 8 = 2 × 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2 × 2
✔️ Final: 64 = 2⁶

🔵 104
🔵 Step 1: 104 is even → divide by 2
🔵 Step 2: 104 = 2 × 52 = 2 × 2 × 26 = 2 × 2 × 2 × 13
✔️ Final: 104 = 2³ × 13

🔵 105
🔵 Step 1: 105 ends with 5 → divisible by 5
🔵 Step 2: 105 = 5 × 21
🔵 Step 3: 21 = 3 × 7
✔️ Final: 105 = 3 × 5 × 7

🔵 243
🔵 Step 1: 2 + 4 + 3 = 9 → divisible by 3
🔵 Step 2: 243 = 3 × 81 = 3 × 3 × 27 = 3 × 3 × 3 × 9 = 3 × 3 × 3 × 3 × 3
✔️ Final: 243 = 3⁵

🔵 320
🔵 Step 1: 320 is even → divide by 2 repeatedly
🔵 Step 2: 320 = 2 × 160 = 2 × 2 × 80 = 2 × 2 × 2 × 40 = 2 × 2 × 2 × 2 × 20 = 2 × 2 × 2 × 2 × 2 × 10 = 2 × 2 × 2 × 2 × 2 × 2 × 5
✔️ Final: 320 = 2⁶ × 5

🔵 141
🔵 Step 1: 1 + 4 + 1 = 6 → divisible by 3
🔵 Step 2: 141 = 3 × 47
✔️ Final: 141 = 3 × 47

🔵 1728
🔵 Step 1: 1728 = 12 × 144
🔵 Step 2: 12 = 2² × 3 and 144 = 2⁴ × 3²
🔵 Step 3: Combine powers
✔️ Final: 1728 = 2⁶ × 3³

🔵 729
🔵 Step 1: 7 + 2 + 9 = 18 → divisible by 3
🔵 Step 2: 729 = 3 × 243 = 3 × 3⁵
✔️ Final: 729 = 3⁶

🔵 1024
🔵 Step 1: Divide repeatedly by 2
🔵 Step 2: 1024 = 2 × 512 = 2 × 2 × 256 = …
✔️ Final: 1024 = 2¹⁰

🔵 1331
🔵 Step 1: 11 × 11 × 11 = 1331
✔️ Final: 1331 = 11³

🔵 1000
🔵 Step 1: 1000 = 10 × 10 × 10
🔵 Step 2: 10 = 2 × 5
✔️ Final: 1000 = 2³ × 5³

🔒 ❓ Question 2.
The prime factorisation of a number has one 2, two 3s, and one 11. What is the number?

📌 ✅ Answer:
🔵 Step 1: Write given factors → 2, 3, 3, 11
🔵 Step 2: Multiply → 2 × 3 × 3 × 11
✔️ Final: 198

🔒 ❓ Question 3.
Find three prime numbers, all less than 30, whose product is 1955.

📌 ✅ Answer:
🔵 Step 1: 1955 ends with 5 → divisible by 5
🔵 Step 2: 1955 = 5 × 391
🔵 Step 3: 391 = 17 × 23
✔️ Final: 5, 17, 23

🔒 ❓ Question 4.
Find the prime factorisation of these numbers without multiplying first.

🔒 ❓ (a) 56 × 25

📌 ✅ Answer:
🔵 Step 1: 56 = 2³ × 7
🔵 Step 2: 25 = 5²
✔️ Final: 56 × 25 = 2³ × 5² × 7

🔒 ❓ (b) 108 × 75

📌 ✅ Answer:
🔵 Step 1: 108 = 2² × 3³
🔵 Step 2: 75 = 3 × 5²
✔️ Final: 108 × 75 = 2² × 3⁴ × 5²

🔒 ❓ (c) 1000 × 81

📌 ✅ Answer:
🔵 Step 1: 1000 = 2³ × 5³
🔵 Step 2: 81 = 3⁴
✔️ Final: 1000 × 81 = 2³ × 3⁴ × 5³

🔒 ❓ Question 5.
What is the smallest number whose prime factorisation has:

🔒 ❓ (a) three different prime numbers?

📌 ✅ Answer:
🔵 Step 1: Take smallest three primes → 2, 3, 5
🔵 Step 2: Multiply → 2 × 3 × 5
✔️ Final: 30

🔒 ❓ (b) four different prime numbers?

📌 ✅ Answer:
🔵 Step 1: Take smallest four primes → 2, 3, 5, 7
🔵 Step 2: Multiply → 2 × 3 × 5 × 7
✔️ Final: 210

🌿 BASED ON PRIME FACTORISATION AND DIVISIBILITY

🔒 ❓ Question 1.
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

📌 ✅ Answer:
🔵 (a) 30 and 45
🔵 Step 1: 30 = 2 * 3 * 5
🔵 Step 2: 45 = 3 * 3 * 5 = 3^2 * 5
🔴 Step 3: Common prime factors = 3 and 5
✔️ Final: Not co-prime

🔵 (b) 57 and 85
🔵 Step 1: 57 = 3 * 19
🔵 Step 2: 85 = 5 * 17
🟡 Step 3: No common prime factor
✔️ Final: Co-prime

🔵 (c) 121 and 1331
🔵 Step 1: 121 = 11 * 11 = 11^2
🔵 Step 2: 1331 = 11 * 11 * 11 = 11^3
🔴 Step 3: Common prime factor = 11
✔️ Final: Not co-prime

🔵 (d) 343 and 216
🔵 Step 1: 343 = 7 * 7 * 7 = 7^3
🔵 Step 2: 216 = 2 * 2 * 2 * 3 * 3 * 3 = 2^3 * 3^3
🟡 Step 3: No common prime factor
✔️ Final: Co-prime

🔒 ❓ Question 2.
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

📌 ✅ Answer:
🔵 (a) 225 and 27
🔵 Step 1: 225 = 3 * 3 * 5 * 5 = 3^2 * 5^2
🔵 Step 2: 27 = 3 * 3 * 3 = 3^3
🔴 Step 3: 225 has only 3^2 but 27 needs 3^3
✔️ Final: No, 225 is not divisible by 27

🔵 (b) 96 and 24
🔵 Step 1: 96 = 2 * 2 * 2 * 2 * 2 * 3 = 2^5 * 3
🔵 Step 2: 24 = 2 * 2 * 2 * 3 = 2^3 * 3
🟢 Step 3: 96 has at least 2^3 * 3 as factors
✔️ Final: Yes, 96 is divisible by 24

🔵 (c) 343 and 17
🔵 Step 1: 343 = 7^3
🔵 Step 2: 17 is a prime number
🔴 Step 3: 17 is not a factor of 343
✔️ Final: No, 343 is not divisible by 17

🔵 (d) 999 and 99
🔵 Step 1: 999 = 3 * 333
🔵 Step 2: 333 = 3 * 111
🔵 Step 3: 111 = 3 * 37
🔵 Step 4: 999 = 3^3 * 37
🔵 Step 5: 99 = 9 * 11 = 3^2 * 11
🔴 Step 6: 999 does not have factor 11
✔️ Final: No, 999 is not divisible by 99

🔒 ❓ Question 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:
🔵 Step 1: First number = 2 * 3 * 7 = 42
🔵 Step 2: Second number = 3 * 7 * 11 = 231
🔴 Step 3: Common prime factors = 3 and 7
✔️ Final: They are not co-prime

🟡 Check (divisibility):
🔵 Step 4: 231 / 42 is not a whole number
🔵 Step 5: 42 / 231 is not a whole number
✔️ Final: Neither number divides the other

🔒 ❓ Question 4.
Guna says, “Any two prime numbers are co-prime?”. Is he right?

📌 ✅ Answer:
🔵 Step 1: Two different prime numbers have no common factor except 1
🟢 Step 2: Example: 3 and 5 are co-prime
🔴 Step 3: But if the two primes are the same, they share that prime as a common factor
🔴 Step 4: Example: 5 and 5 have common factor 5
✔️ Final: No, he is not right (only two different prime numbers are co-prime)

🌿 BASED ON DIVISIBILITY

🔒 ❓ Question 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:
🔵 Step 1: A leap year is divisible by 4
🔵 Step 2: Years divisible by 100 are leap years only if divisible by 400
🟡 Teacher’s note:
This depends on the student’s year of birth. List all years from your birth year till now that satisfy the rule above.
✔️ Final: Answer will vary from student to student

🔒 ❓ (b) From the year 2024 till 2099, how many leap years are there?

📌 ✅ Answer:
🔵 Step 1: First leap year = 2024
🔵 Step 2: Last leap year before 2100 = 2096
🔵 Step 3: Count with step 4
(2096 − 2024) / 4 + 1
= 72 / 4 + 1
= 18 + 1
✔️ Final: 19 leap years

🔒 ❓ Question 2.
Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.

📌 ✅ Answer:
🔵 Step 1: A 4-digit palindrome has the form abba
🔵 Step 2: A number is divisible by 4 if its last two digits are divisible by 4

🔵 Smallest check:
1001 → 01 ❌
1221 → 21 ❌
1441 → 41 ❌
1661 → 61 ❌
1881 → 81 ❌
2112 → 12 ✔️

🔵 Largest check:
9999 → 99 ❌
9889 → 89 ❌
9669 → 69 ❌
9449 → 49 ❌
9229 → 29 ❌
9009 → 09 ❌
8888 → 88 ✔️

✔️ Final:
Smallest = 2112
Largest = 8888

🔒 ❓ Question 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:
🔵 Example: 2 + 6 = 8 ✔️
🔴 Example: 2 + 4 = 6 ❌
✔️ Final: Sometimes true

🔒 ❓ (b) Sum of two odd numbers gives a multiple of 4.

📌 ✅ Answer:
🔵 Example: 1 + 3 = 4
🔵 Example: 5 + 7 = 12
✔️ Final: Always true

🔒 ❓ Question 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

📌 ✅ Answer:
🔵 Divided by 10 (remainder = last digit)
78 → 8
99 → 9
173 → 3
572 → 2
980 → 0
1111 → 1
2345 → 5

🔵 Divided by 5
78 → 3
99 → 4
173 → 3
572 → 2
980 → 0
1111 → 1
2345 → 0

🔵 Divided by 2
78 → 0
99 → 1
173 → 1
572 → 0
980 → 0
1111 → 1
2345 → 1

✔️ Final: Remainders listed correctly

🔒 ❓ Question 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:
🔵 Step 1: Divisible by 8 ⇒ divisible by 2 and 4
🔵 Step 2: Divisible by 10 ⇒ divisible by 2 and 5
✔️ Final: 8 and 10

🔒 ❓ Question 6.
Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10:
572, 2352, 5600, 6000, 77622160

📌 ✅ Answer:
🔵 Step 1: LCM of 2, 4, 5, 8, 10 = 40
🔵 Step 2: Check divisibility by 40
572 ❌
2352 ❌
5600 ✔️
6000 ✔️
77622160 ✔️
✔️ Final: 5600, 6000, 77622160

🔒 ❓ Question 7.
Write two numbers whose product is 10000. The two numbers should not have 0 as the units digit.

📌 ✅ Answer:
🔵 Step 1: 10000 = 2⁴ × 5⁴
🔵 Step 2: Choose factors not ending in 0
🔵 Example: 16 × 625 = 10000
✔️ Final: 16 and 625

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OTHER IMPORTANT QUESTIONS

(CBSE MODEL QUESTION PAPER)

ESPECIALLY MADE FROM THIS CHAPTER ONLY

🔵 Section A — Very Short Answer (1 mark each)

🔒 ❓ Question 1
What is a factor of a number?
📌 ✅ Answer:
🔹 A factor is a number that divides another number exactly
🔸 It leaves no remainder

🔒 ❓ Question 2
Write any one factor of 15.
📌 ✅ Answer:
🔹 One factor of 15 is 3

🔒 ❓ Question 3
What is a multiple of a number?
📌 ✅ Answer:
🔹 A multiple is obtained by multiplying a number by a whole number

🔒 ❓ Question 4
Write the smallest prime number.
📌 ✅ Answer:
🔹 The smallest prime number is 2

🔒 ❓ Question 5
Is 1 a prime number?
📌 ✅ Answer:
🔹 1 has only one factor
✔️ Final: No

🔒 ❓ Question 6
True or False:
Every even number is a prime number.
📌 ✅ Answer:
🔹 Only 2 is an even prime number
✔️ Final: False

🟢 Section B — Short Answer I (2 marks each)

🔒 ❓ Question 7
Write any two prime numbers.
📌 ✅ Answer:
🔹 3 and 7 are prime numbers

🔒 ❓ Question 8
Write any two composite numbers.
📌 ✅ Answer:
🔹 4 and 9 are composite numbers

🔒 ❓ Question 9
Find the factors of 12.
📌 ✅ Answer:
🔹 1, 2, 3, 4, 6, and 12 are factors of 12

🔒 ❓ Question 10
Write the first four multiples of 6.
📌 ✅ Answer:
🔹 6, 12, 18, and 24 are multiples of 6

🔒 ❓ Question 11
Why is 2 called a prime number?
📌 ✅ Answer:
🔹 2 has exactly two factors: 1 and 2
🔸 Therefore, it is a prime number

🔒 ❓ Question 12
What are co-prime numbers?
📌 ✅ Answer:
🔹 Co-prime numbers have only one common factor
🔸 Their only common factor is 1

🟡 Section C — Short Answer II (3 marks each)

🔒 ❓ Question 13
Explain the difference between factors and multiples.

📌 ✅ Answer:
🔹 Factors are numbers that divide a given number exactly
🔹 Multiples are obtained by multiplying a number by whole numbers
🔸 A number has limited factors but unlimited multiples

🔒 ❓ Question 14
Find all the factors of 20.

📌 ✅ Answer:
🔹 1 × 20 = 20
🔹 2 × 10 = 20
🔹 4 × 5 = 20
🔸 Factors of 20 are 1, 2, 4, 5, 10, 20

🔒 ❓ Question 15
Write the first five multiples of 7.

📌 ✅ Answer:
🔹 Multiples of 7 are obtained by multiplying 7 by 1, 2, 3, 4, 5
🔸 First five multiples are 7, 14, 21, 28, 35

🔒 ❓ Question 16
Why is 1 neither a prime number nor a composite number?

📌 ✅ Answer:
🔹 Prime numbers have exactly two factors
🔹 Composite numbers have more than two factors
🔸 Number 1 has only one factor, so it is neither prime nor composite

🔒 ❓ Question 17
Explain why 9 is a composite number.

📌 ✅ Answer:
🔹 Factors of 9 are 1, 3, and 9
🔹 It has more than two factors
🔸 Therefore, 9 is a composite number

🔒 ❓ Question 18
Write two pairs of co-prime numbers.

📌 ✅ Answer:
🔹 (8, 15) are co-prime because their only common factor is 1
🔸 (9, 20) are co-prime because they have no common factor other than 1

🔒 ❓ Question 19
State the divisibility rule of 2 and 5.

📌 ✅ Answer:
🔹 A number is divisible by 2 if its last digit is even
🔸 A number is divisible by 5 if its last digit is 0 or 5

🔒 ❓ Question 20
Check whether 135 is divisible by 3. Give reason.

📌 ✅ Answer:
🔹 Sum of digits of 135 = 1 + 3 + 5 = 9
🔹 Since 9 is divisible by 3
🔸 Therefore, 135 is divisible by 3

🔒 ❓ Question 21
How many prime numbers are there between 1 and 20? Name them.

📌 ✅ Answer:
🔹 Prime numbers between 1 and 20 are 2, 3, 5, 7, 11, 13, 17, 19
🔸 There are 8 prime numbers

🔒 ❓ Question 22
Why are divisibility rules useful?

📌 ✅ Answer:
🔹 Divisibility rules help check division quickly
🔹 They save time and effort in calculations
🔸 They reduce chances of calculation mistakes

🔴 Section D — Long Answer (4 marks each)

🔒 ❓ Question 23
Explain how factors of a number can be found. Illustrate with an example.

📌 ✅ Answer:
🔹 To find factors, divide the number by whole numbers starting from 1
🔹 If the division leaves no remainder, the divisor is a factor
🔹 Factors occur in pairs
🔸 Example: For 24 → (1, 24), (2, 12), (3, 8), (4, 6)
✔️ Final: Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24

🔒 ❓ Question 24
Explain the difference between prime numbers and composite numbers with examples.

📌 ✅ Answer:
🔹 Prime numbers have exactly two factors: 1 and the number itself
🔹 Example: 7 has factors 1 and 7
🔹 Composite numbers have more than two factors
🔸 Example: 12 has factors 1, 2, 3, 4, 6, 12
✔️ Final: Primes have two factors; composites have more than two

🔒 ❓ Question 25
Why is 2 the only even prime number? Explain.

📌 ✅ Answer:
🔹 Every even number greater than 2 is divisible by 2
🔹 Such numbers have more than two factors
🔹 Number 2 has only two factors: 1 and 2
🔸 Therefore, 2 is prime and no other even number is prime
✔️ Final: 2 is the only even prime number

🔒 ❓ Question 26
List the prime numbers between 1 and 50 and state how many there are.

📌 ✅ Answer:
🔹 Prime numbers between 1 and 50 are:
🔹 2, 3, 5, 7, 11, 13, 17, 19
🔹 23, 29, 31, 37, 41, 43, 47
🔸 Total number of prime numbers = 15
✔️ Final: There are 15 prime numbers between 1 and 50

🔒 ❓ Question 27
OR
Explain what co-prime numbers are with two examples.

📌 ✅ Answer:
🔹 Co-prime numbers have only one common factor, which is 1
🔹 Example 1: 14 and 25 (no common factor other than 1)
🔸 Example 2: 9 and 20 (no common factor other than 1)
✔️ Final: Co-prime numbers share only factor 1

🔒 ❓ Question 28
Explain the importance of divisibility rules with suitable examples.

📌 ✅ Answer:
🔹 Divisibility rules help check division quickly without long calculations
🔹 Example: 246 is divisible by 2 because the last digit is even
🔹 Example: 405 is divisible by 5 because the last digit is 5
🔸 They save time and reduce calculation errors
✔️ Final: Divisibility rules make calculations faster and easier

🔒 ❓ Question 29
Explain how prime numbers help in understanding other numbers.

📌 ✅ Answer:
🔹 Every composite number can be expressed using prime factors
🔹 Prime numbers are building blocks of numbers
🔹 They help in finding LCM and HCF
🔸 They are useful in higher mathematics
✔️ Final: Prime numbers form the base of number system

🔒 ❓ Question 30
OR
Explain how knowledge of factors and multiples is useful in daily life.

📌 ✅ Answer:
🔹 Factors help in equal sharing of items
🔹 Multiples help in planning repeated events
🔹 Example: Arranging students in equal rows
🔸 Example: Scheduling activities at fixed intervals
✔️ Final: Factors and multiples make daily planning easier

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