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SecA

Using Divisibility rules find:

1. A number is divisible by both 5 and 9. Is it divisible by 45? Justify your answer.

2. Determine if 57 is a prime number.

Sol

Solutions:

1. Yes, it is divisible by 45 because 45 is the least common multiple (LCM) of 5 and 9.

2. No, 57 is not a prime number because it is divisible by 3 (57 = 3 × 19).

SecB

Find the GCD of 48 and 64 using the prime factorization method.

Sol

Solutions:

Prime factors of 48: 24 x 3

Prime factors of 64: 26

GCD: 24 = 16

SecC

1.Two friends, Alice and Bob, are planting trees in their gardens. Alice plants a tree every 3 days, and Bob plants a tree every 5 days. If they both plant a tree on the same day, after how many days will they both plant a tree again on the same day?

2. A baker bakes 120 muffins and 90 cookies. She wants to pack them into boxes such that each box contains an equal number of muffins and an equal number of cookies. What is the greatest number of boxes she can use?

3. Sam has 36 apples and 60 oranges. He wants to pack them in boxes such that each box contains the same number of fruits and there are no fruits left. What is the greatest number of fruits he can pack in each box?

Sol

Solutions:

1. They will both plant a tree again on the same day after the LCM of 3 and 5, which is 15 days.

2. The greatest number of boxes she can use is the GCD of 120 and 90, which is 30.

3. The greatest number of fruits he can pack in each box is the GCD of 36 and 60, which is 12.

SecD

Pranay asks Shruthi to choose a three-digit number. He then asks her to form two more three-digits number keeping the order of same i.e either clockwise or anti-clockwise. He then asks her to add them up and then divide the resulting answer by 37. He guarantees that there will be no remainder. What is the logic behind his guarantee?

Sol

Solutions:

Let the three-digit number be represented as 100A + 10B + C

Form two more three-digit numbers by rotating the digits:

One rotation: BCA, represented as 100B + 10C +A

Another rotation: CAB, represented as 100C + 10A +B

The sum of the three numbers:

(100A + 10B + C) + (100B + 10C +A) + (100C + 10A +B)

Combine like terms:

100A + 10B + C + 100B + 10C +A + 100C + 10A +B

= (100A + A + 10A) + (10B + 100B +B) + (C + 10C + 100C)

= 111A + 111B +111C = 111(A + B + C)

Divide the resulting sum by 37:

Since 111 is a multiple of 37 (111 = 37 X 3):

= = 111(A + B + C) = 37 X 3 (A + B + C)

The sum 111(𝐴+𝐵+𝐶) is always divisible by 37 because 111 is a multiple of 37.

Therefore, dividing 111(𝐴+𝐵+𝐶) by 37 will always result in an integer, guaranteeing no remainder.

P1

Understanding Equality and Fairness:

Question:

Ravi and Priya are playing a game where they roll a pair of dice and add the numbers. If the sum is an even number, Ravi gets a point; if it's an odd number, Priya gets a point. After 10 rounds, the sums of the dice were: 4, 7, 9, 6, 8, 11, 3, 10, 5, and 12. Calculate the points each player has. What value does Ravi demonstrate if he congratulates Priya and asks her to teach him her strategy, instead of feeling upset or accusing her of cheating?

Sol

Solution

To calculate the points:

• Even sums: 4, 6, 8, 10, 12 (Ravi gets 5 points)

• Odd sums: 7, 9, 11, 3, 5 (Priya gets 5 points) Ravi and Priya both end up with 5 points each. Value Learned: If Ravi congratulates Priya and asks her to teach him her strategy, he demonstrates sportsmanship, humility, and a willingness to learn. He values fairness and recognizes Priya’s success, showing respect and a positive attitude towards learning from others.

P2

Integrity in Reporting Scores:

Question:

During a math competition, students are asked to report the number of prime numbers between 1 and 50. Ritu finds 15 prime numbers, which are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Her friend mistakenly counts 16 and insists on reporting the higher number to win a prize. What values should Ritu uphold in this situation, and why?

Sol

Solution

The prime numbers between 1 and 50 are correctly listed by Ritu. Her count is accurate at 15. Value Learned: Ritu should uphold honesty and integrity by reporting the correct number of prime numbers, even if it means not winning the prize. This demonstrates the importance of truthfulness and maintaining ethical standards in competitions and life.

P3

Collaboration and Teamwork:

Question:

A group of students is given a task to find the least common multiple (LCM) of 12, 18, and 24. They decide to divide the task among themselves: one student lists the multiples of each number, another finds the common multiples, and a third identifies the smallest common multiple. What values are reflected in their approach to solving the problem?

Sol

Solution

To find the LCM:

Multiples of 12: 12, 24, 36, 48, 60, 72, ...

Multiples of 18: 18, 36, 54, 72, ...

Multiples of 24: 24, 48, 72, ... The common multiples are 72 (smallest). Value Learned: The students' approach reflects collaboration, teamwork, and effective communication. By dividing the task, they promote cooperation and ensure everyone contributes to solving the problem efficiently, highlighting the importance of working together towards a common goal.

P4

Question:

Two friends, Rahul and Anya, are participating in a running race around a circular track. Rahul completes one round in 8 minutes, while Anya completes one round in 12 minutes. If they both start running at the same time, after how many minutes will they meet again at the starting point? Discuss the values of perseverance and friendly competition demonstrated by Rahul and Anya as they continue to run together.

Sol

Solution

To find when Rahul and Anya will meet again at the starting point, we need to find the least common multiple (LCM) of their round times.

Prime factorization:

8 = 2³

12 = 2² × 3

The LCM of 8 and 12 is 23 × 3 = 242³ × 3 = 2423 × 3 =24.

Therefore, Rahul and Anya will meet again at the starting point after 24 minutes. Value Learned: Rahul and Anya demonstrate perseverance as they continue running and eventually meet at the starting point again. This shows their determination to keep going despite the challenge of different paces. The scenario also highlights friendly competition, as they motivate each other to maintain their efforts and celebrate meeting at the starting point, reinforcing the importance of support and encouragement in achieving personal goals.

Q1

The difference between a two digit number and the number obtained by interchanging its digits is 63. What is the difference between the two digits of the number?

Sol

Solution:

Let the numbers be x and y.

The number at ones place is y and number on tens place is 10x

so the digit will be 10x + y

The interchange of digit is = 10y + x

It is given the difference between the digit and interchange is 63

so 10x + y - (10y + x) = 63

9x - 9y = 63

x - y = 63

Therfore The difference between the digits is 7.

Q2

2. A two digit number exceeds the sum of digits of that number by 18. If the digits at the unit's place is double the digit in the ten's place, find the number

Sol

Solution

Let the number be = 10a + b

10a + b = a + b + 18

10a = a + 18

9a = 18

a = 2

2a = b => b = 4

Q3

3. In a two digit number the digit in the one's place is three times the digit in the ten's place and the sum of the digits is equal to 12. what is the number?

Sol

Solution

Let's denote the two-digit number as 10a + b, where a is the digit in the ten's place and b is the digit in the one's place.

Conditions:

1. The digit in the one's place (b) is three times the digit in the ten's place (a).

2. The sum of the digits is equal to 12.

We can write these conditions as equations:

1. b = 3a

2. a + b = 12

Now, substitute the value of b from the first equation into the second equation:

a + 3a = 12

4a = 12 => a = 3

Now, use the value of a to find b:

b = 3a

b = 3 x 3 => b = 9

So, the digits are a = 3 and b = 9

Therefore, the two-digit number is:

10a + b = 10 x 3 + 9 = 30 + 9 = 39

Hence, the number is 39.

Q4

4. Check the divisibility of 2147681 by 3

Sol

Solution

2 + 1 + 4 + 7 + 6 + 8 + 1 =

which is not divisible by .

Therefore, 2147681 is not divisible by 3

Questions

1. A number is divisible by 5 and 6. It may not be divisible by

2. The sum of the prime factors of 1729 is

3. The number of common prime factors of 75, 60, 105 is

4. Sum of two consecutive odd numbers is always divisible by 4.

5. A number with 4 or more digits is divisible by 8, if the number formed by the last three digits is divisible by 8.

6. The Highest Common Factor of two or more numbers is greater than their Lowest Common Multiple.

7. If the HCF of two numbers is one of the numbers, then their LCM is the other number.

8. A number for which the sum of all its factors is equal to twice the number is called a number.

9. Two numbers having only 1 as a common factor are called_numbers.

10. If the difference between the sum of digits at odd places (from the right) and the sum of digits at even places (from the right) of a number is either 0 or divisible by , then the number is divisible by 11.

11. A number is divisible by if it has any of the digits 0, 2, 4, 6, or 8 in its ones place.

12. In a five digit number, digit at ten’s place is 4, digit at unit’s place is one fourth of ten’s place digit, digit at hunderd’s place is 0, digit at thousand’s place is 5 times of the digit at unit’s place and ten thousand’s place digit is double the digit at ten’s place. Write the number.

13. A factory has a container filled with 35874 litres of cold drink. In how many bottles of 200 ml capacity each can it be filled?

14. Find the LCM of 80, 96, 125, 160.

15. A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of a third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

Sol

Solutions

1. 60

2. 39

3. 3 and 5

4. True

5. True

6. False

7. True

8. Perfect

9. co-primes

10. 11

11. 2

12. 85041

13. 179370

14. 12000

15. 60 liters.

Q1

A florist has 200 roses, 180 marigolds, and 320 orchids. He needs to make garlands, each containing the same number of flowers but using only one type of flower per garland (either only roses, only marigolds, or only orchids).

Based on the above information, answer the following questions

(i)What is the Highest Common Factor (HCF) of two co-prime numbers?

(ii) What is the Least Common Multiple (LCM) of two co-prime numbers?

(iii)Find the LCM of 200 and 320

Sol

Solutions

(i) The HCF of two co-prime numbers is always 1. This is because co-prime numbers have no common factors other than 1.

(ii) The LCM of two co-prime numbers is the product of the numbers themselves. This is because, having no common factors other than 1, the smallest number that is a multiple of both is their product.

(iii)

1. Prime Factorization:

• 200: 200 = 23 x 5232

• 320: 320 = 26 x 51

2. LCM Calculation:

The LCM is found by taking the highest power of each prime factor that appears in the factorizations.

LCM = 2max3,6x5max2,1

LCM = 26 x 52

LCM = 64 x 25 => LCM = 16001420.

So, the LCM of 200 and 320 is 1600.

Q2

In a classroom activity called "The Mystery Boxes," each student receives a box with a different number inside. The students have to solve problems based on the number in their box. Sarah's box contains the number 48, Tom's box contains the number 30, and Zoe's box contains the number 23.

Based on the above information, answer the following questions

(i)List all the factors of Sarah's number?

(ii) Is Tom's number a multiple of 5? Explain your reasoning.

(iii)Is Sarah's number a prime or composite number? Explain.

(iv)Determine if Zoe's number is prime or composite.

(v)Write down the prime factors of Tom's number?

(vi)What is the greatest common divisor (GCD) of Tom's number and Zoe's number?

(vii)Find the least common multiple (LCM) of Sarah's number and Tom's number?

Sol

Solutions

(i) The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

(ii) Yes, 30 is a multiple of 5 because 30 ÷ 5 = which is a whole number.

(iii) 48 is a composite number because it has more than two factors.

(iv) 23 is a prime numberperfecr number because its only factors are 1 and 23.

(v) The prime factors of 30 are 2, 3, and (since 30 = 2 × 3 × 5)

(vi) The GCD of 30 and 23 is 1, since 23 is a prime number and does not share any common factors with 30.

(vii) The LCM of 48 and 30 is 240120 (since 48 = 24 × 3 and 30 = 2 × 3 × 5, the LCM is 24 × 3 × 5 = 16 × 3 × 5 = 240)