Question 1

Q1: List all the factors of the following numbers: (a) 90 (b) 105 (c) 132 (d) 360 (this number has 24 factors) (e) 840 (this number has 32 factors)

Accepted Answer

To Find: All factors of the given numbers. Solution: We find the prime factorisation first and then list all possible combinations of these prime factors (including 1). (a) 90 Prime factorisation: $90 = 2 	imes 3 	imes 3 	imes 5 = 2^1 	imes 3^2 	imes 5^1$. Factors are formed by taking combinati...

Question 2

Q1: Sameeksha is building her new house．The main room of the house is 12 ft by 16 ft ．She feels that the room would look nice if the floor is covered with...

Accepted Answer

Given: - Dimensions of the room: 12 ft by 16 ft. - The floor is to be covered with square tiles of the same size. - The side length of the tile must be a whole number of feet. - The number of tiles used should be as few as possible. To Find: The size of the tiles Sameeksha should buy. Solution: For...

Question 3

Q2: How many tiles of this size should she purchase?

Accepted Answer

Given: - Dimensions of the room: 12 ft by 16 ft. - Size of one square tile: 4 ft by 4 ft. To Find: The number of tiles required to cover the floor. Solution: Area of the room = Length $	imes$ Breadth = $16 	ext{ ft} 	imes 12 	ext{ ft} = 192 	ext{ sq ft}$. Area of one tile = Side $	imes$ Side =...

Question 4

Q3: What if Sameeksha did not insist on the length of the tile to be a whole number of feet and the length could be a fractional number of feet? Would the...

Accepted Answer

Given: - The length of the tile can be a fractional number of feet. To Find: - Whether the answer for the largest tile size would change. Solution: If the length of the tile could be a fractional number, the problem remains the same: to find the largest number that can divide both 12 and 16 exactly....

Question 5

Q4: Lekhana purchases rice from two farms and sells it in the market. She bought 84 kg of rice from one farm and 108 kg from the other farm. She wants the...

Accepted Answer

Given: - Weight of rice from the first farm: 84 kg. - Weight of rice from the second farm: 108 kg. - The rice is to be packed in bags of equal weight. - The number of bags should be as few as possible. To Find: The weight of each bag. Solution: To use as few bags as possible, the weight of each bag...

Question 6

Q5: In each case below, the two numbers upon which treasures are kept are given. Find the longest jump size (starting from 0) using which Jumpy can land o...

Accepted Answer

Given: Pairs of numbers where treasures are kept. To Find: The longest jump size to land on both numbers in each pair. This is equivalent to finding the Highest Common Factor (HCF) of each pair. Solution: (a) 14 and 30 Factors of 14: 1, 2, 7, 14 Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Common factor...

Question 7

Q6: Is the longest jump size for the numbers the same as their HCF? Explain why it is so.

Accepted Answer

To Find: Whether the longest jump size is the same as the HCF and explain why. Solution: Yes, the longest jump size is the same as the HCF of the two numbers. Explanation: For Jumpy to land on a number (say 'a') starting from 0 with a jump size of 'j', the number 'a' must be a multiple of the jump s...

Question 8

Q1: Can you write the prime factorisation of 105 and 30 using these two figures?

Accepted Answer

To Find: The prime factorisation of 105 and 30. Solution: We will use the division method for prime factorisation. For 105: We start by dividing by the smallest prime factor, which is 3. $105 \div 3 = 35$ Now, we find the smallest prime factor of 35, which is 5. $35 \div 5 = 7$ 7 is a prime number,...

Question 9

Q2: Try finding the prime factorisation of 1200 using the method above.

Accepted Answer

To Find: The prime factorisation of 1200. Solution: Using the division method: $1200 \div 2 = 600$ $600 \div 2 = 300$ $300 \div 2 = 150$ $150 \div 2 = 75$ $75 \div 3 = 25$ $25 \div 5 = 5$ $5 \div 5 = 1$ The prime factors are 2, 2, 2, 2, 3, 5, 5. Final Answer: The prime factorisation of 1200 is $2 	...

Question 10

Q3: Is $2 	imes 2 	imes 7=28$ a factor of $840$?

Accepted Answer

Given: The number 840 and a potential factor 28. Prime factorisation of 840 is $2 	imes 2 	imes 2 	imes 3 	imes 5 	imes 7$. To Find: Whether 28 is a factor of 840. Solution: A number is a factor if its prime factors are a 'subpart' of the prime factorisation of the larger number. The prime fact...

Question 11

Q4: If yes, what should it be multiplied by to get 840?

Accepted Answer

Given: $840 = (2 	imes 2 	imes 7) 	imes (2 	imes 3 	imes 5)$ To Find: The number that 28 ($2 	imes 2 	imes 7$) must be multiplied by to get 840. Solution: From the rearranged prime factorisation, we can see that the remaining factors are $2 	imes 3 	imes 5$. $2 	imes 3 	imes 5 = 30$. So,...

Question 12

Q5: Similarly, is $2 	imes 7=14$ a factor of 840? Why or why not?

Accepted Answer

To Find: Whether 14 is a factor of 840 and why. Solution: Yes, 14 is a factor of 840. Reason: The prime factorisation of 14 is $2 	imes 7$. The prime factorisation of 840 is $2 	imes 2 	imes 2 	imes 3 	imes 5 	imes 7$. The prime factors of 14 ($2, 7$) are present in the prime factorisation of...

Question 13

Q6: Is $2 	imes 2 	imes 2$ a factor of 840? Why or why not?

Accepted Answer

To Find: Whether $2 	imes 2 	imes 2 = 8$ is a factor of 840 and why. Solution: Yes, 8 is a factor of 840. Reason: The prime factorisation of 8 is $2 	imes 2 	imes 2$. The prime factorisation of 840 is $2 	imes 2 	imes 2 	imes 3 	imes 5 	imes 7$. The prime factors of 8 ($2, 2, 2$) are presen...

Question 14

Q7: Is $3 	imes 3 	imes 3$ a factor of 840? Why or why not?

Accepted Answer

To Find: Whether $3 	imes 3 	imes 3 = 27$ is a factor of 840 and why. Solution: No, 27 is not a factor of 840. Reason: The prime factorisation of 27 is $3 	imes 3 	imes 3$. The prime factorisation of 840 is $2 	imes 2 	imes 2 	imes 3 	imes 5 	imes 7$. The prime factorisation of 840 contains...

NCERT Solutions