Finding Common Ground
1. What is the Highest Common Factor (HCF) of 12 and 16? Explain using factors.
Answer:
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 16 = 1, 2, 4, 8, 16
Common factors = 1, 2, 4
The greatest common factor is 4.
HCF = 4
Explanation: The HCF is the largest number that divides both numbers exactly. Since 4 is the largest common factor of 12 and 16, their HCF is 4.
2. Sameeksha wants to cover a room of size 12 ft × 16 ft using the largest square tiles possible. What should be the side length of each tile?
Answer:
The tile side must divide both 12 and 16 exactly.
HCF of 12 and 16 = 4
Tile side = 4 ft
Explanation: A square tile of side 4 ft fits exactly along both the length and breadth of the room. Using the largest tile reduces the total number of tiles needed.
3. Lekhana has 84 kg and 108 kg of rice. She wants bags of equal weight and wants to use the fewest bags possible. What should be the weight of each bag?
Answer:
Prime factors:
84 = 2 × 2 × 3 × 7
108 = 2 × 2 × 3 × 3 × 3
Common prime factors = 2 × 2 × 3
HCF = 12
Weight of each bag = 12 kg
Explanation: The largest common factor gives the largest possible bag size. Larger bags mean fewer bags are needed.
4. What is a prime number? Give two examples from the chapter.
Answer:
A prime number is a number greater than 1 that has only two factors:
- 1
- The number itself
Examples: 3, 5, 7, 11
Explanation: Prime numbers cannot be divided exactly by any other number except 1 and themselves.
5. Find the prime factorisation of 105 using the division method.
Answer:
105 ÷ 3 = 35
35 ÷ 5 = 7
7 ÷ 7 = 1
Therefore,
105 = 3 × 5 × 7
Explanation: We keep dividing by prime numbers until only 1 remains. The primes obtained form the prime factorisation.
6. Find the prime factorisation of 30.
Answer:
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
Therefore,
30 = 2 × 3 × 5
Explanation: The number 30 is expressed completely as a product of prime numbers.
7. Why does every method of factorising 90 lead to the same prime factors?
Answer:
For example,
90 = 3 × 30
90 = 3 × 2 × 15
90 = 3 × 2 × 3 × 5
Prime factors = 2, 3, 3, 5
Explanation: Although the order of factorisation may change, the final prime factors remain the same. Only their arrangement changes.
8. Using prime factorisation, list all factors of 225.
Answer:
Prime factorisation:
225 = 3 × 3 × 5 × 5
Factors:
1, 3, 5, 9, 15, 25, 45, 75, 225
Explanation: Factors are obtained by forming different combinations of the prime factors.
9. Anshu says, “The larger a number is, the longer its prime factorisation will be.” Is he correct?
Answer:
No.
Example:
96 = 2 × 2 × 2 × 2 × 2 × 3
121 = 11 × 11
Explanation: 121 is larger than 96, but its prime factorisation is shorter. Therefore, Anshu’s statement is false.
10. What is a conjecture?
Answer:
A conjecture is a statement or claim made without proof or verification.
Explanation:
In mathematics, a conjecture must be tested or proved before it can be accepted as true.
11. Find the HCF of 45 and 75 using prime factorisation.
Answer:
45 = 3 × 3 × 5
75 = 3 × 5 × 5
Common primes = 3 × 5
HCF = 15
Explanation: The HCF contains the common prime factors with the smallest number of occurrences.
12. Find the HCF of 112 and 84.
Answer:
112 = 2 × 2 × 2 × 2 × 7
84 = 2 × 2 × 3 × 7
Common primes = 2 × 2 × 7
HCF = 28
Explanation: The common prime factors are multiplied to obtain the highest common factor.
13. Why is the HCF of 96 and 275 equal to 1?
Answer:
96 = 2 × 2 × 2 × 2 × 2 × 3
275 = 5 × 5 × 11
There are no common prime factors.
Therefore,
HCF = 1
Explanation: When two numbers have no common prime factor, 1 is their only common factor.
14. Anshu uses strips of 6 cm and Guna uses strips of 8 cm. What is the shortest toran length both can make?
Answer:
Multiples of 6:
6, 12, 18, 24, …
Multiples of 8:
8, 16, 24, …
Smallest common multiple = 24
LCM = 24 cm
Explanation: The shortest common length is the least common multiple of 6 and 8.
15. Kabamai visits a sweet shop every 10 days. Free sweets are given every 7 days. After how many days will she get free sweets again?
Answer:
Multiples of 7:
7, 14, 21, 28, 35, 42, 49, 56, 63, 70
Multiples of 10:
10, 20, 30, 40, 50, 60, 70
First common multiple = 70
Answer = 70 days
Explanation: The first day when both events occur together is the LCM of 7 and 10.
16. Find the LCM of 14 and 35 using prime factorisation.
Answer:
14 = 2 × 7
35 = 5 × 7
Take all prime factors:
LCM = 2 × 5 × 7
LCM = 70
Explanation: For the LCM, include each prime factor the maximum number of times it appears.
17. Find the LCM of 96 and 360.
Answer:
96 = 2 × 2 × 2 × 2 × 2 × 3
360 = 2 × 2 × 2 × 3 × 3 × 5
Take highest powers:
LCM = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5
= 32 × 9 × 5
= 1440
LCM = 1440
Explanation: The LCM must contain every prime factor needed to form both numbers.
18. When is the HCF of two numbers equal to one of the given numbers?
Answer:
When one number is a factor of the other.
Example:
HCF of 6 and 18 = 6
Explanation: Since 6 divides 18 exactly, the smaller number becomes the HCF.
19. What happens to the HCF if both numbers are doubled?
Answer:
The HCF also doubles.
Example:
270 and 50
HCF = 10
After doubling:
540 and 100
HCF = 20
Explanation: Both numbers gain an extra factor of 2, so the HCF also gains an extra factor of 2.
20. What is the relationship between HCF, LCM, and the product of two numbers?
Answer:
HCF × LCM = Product of the two numbers
Example:
105 = 3 × 5 × 7
95 = 5 × 19
HCF = 5
LCM = 3 × 5 × 7 × 19 = 1995
HCF × LCM
= 5 × 1995
= 9975
Product of numbers
= 105 × 95
= 9975
Therefore,
HCF × LCM = 105 × 95
Explanation: For any two numbers, the product of their HCF and LCM is equal to the product of the numbers themselves. This is an important property discussed in the chapter.
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