Answer :
Let's solve this problem step-by-step.
### 1. Greatest Common Factor (GCF) of 36 and 60
To find the greatest number that divides both 36 and 60 exactly, we need to find their Greatest Common Factor (GCF).
#### Step-by-Step:
1. Prime factorization of 36:
- 36 = 2 × 2 × 3 × 3 = 2² × 3²
2. Prime factorization of 60:
- 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
3. Common factors:
- Both 36 and 60 have the factors: 2² (which is 4) and 3.
4. Product of the common factors:
- The common factors are 2² and 3.
- Thus, GCF = 2² × 3 = 4 × 3 = 12
So, the greatest number that divides both 36 and 60 exactly is 12.
### 2. Prime Factorization and Common Factors of 54 and 90
Let's address each part:
#### (i) Express each number as the product of their prime factors:
1. Prime factorization of 54:
- 54 = 2 × 3 × 3 × 3 = 2 × 3³
2. Prime factorization of 90:
- 90 = 2 × 3 × 3 × 5 = 2 × 3² × 5
#### (ii) Find the product of the common prime factors of these numbers:
1. Common prime factors of 54 and 90:
- Both have the factors 2 and 3.
- The highest power of common factor 3 is 3².
2. Product of common prime factors:
- Product = 2 × 3² = 2 × 9 = 18
#### (iii) Is the product the H.C.F. of the numbers?
1. HCF (Highest Common Factor) Calculation:
- The common factors identified above (2 and 3²) already confirm the HCF calculation directly.
2. Why is this the HCF?
- The HCF is the largest number that divides both 54 and 90 without a remainder.
- The prime factorizations of 54 and 90 share the factor 2 and 3².
3. As determined, the product of these common factors is indeed 18, so Yes, the product is the HCF of the numbers.
### 3. Distribution of 24 Snickers and 36 Cadburies
Bishwant distributes these equally among his friends. We need to find suitable numbers of friends.
#### (i) Possible numbers of friends to whom the snickers can be equally distributed:
1. Factors of 24:
- 1, 2, 3, 4, 6, 8, 12, 24
#### (ii) Possible numbers of friends to whom the cadburies can be equally distributed:
1. Factors of 36:
- 1, 2, 3, 4, 6, 9, 12, 18, 36
#### (iii) Possible numbers of friends to whom both the snickers and cadburies can be equally distributed:
1. Common factors between 24 and 36:
- Common factors = 1, 2, 3, 4, 6, 12
#### (iv) Greatest number of friends to whom the candies can be distributed equally:
1. Greatest common divisor (GCD) of 24 and 36:
- From the common factors, the greatest number is 12.
So, the greatest number of friends to whom the 24 snickers and 36 cadburies can be distributed equally is 12.
### 1. Greatest Common Factor (GCF) of 36 and 60
To find the greatest number that divides both 36 and 60 exactly, we need to find their Greatest Common Factor (GCF).
#### Step-by-Step:
1. Prime factorization of 36:
- 36 = 2 × 2 × 3 × 3 = 2² × 3²
2. Prime factorization of 60:
- 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
3. Common factors:
- Both 36 and 60 have the factors: 2² (which is 4) and 3.
4. Product of the common factors:
- The common factors are 2² and 3.
- Thus, GCF = 2² × 3 = 4 × 3 = 12
So, the greatest number that divides both 36 and 60 exactly is 12.
### 2. Prime Factorization and Common Factors of 54 and 90
Let's address each part:
#### (i) Express each number as the product of their prime factors:
1. Prime factorization of 54:
- 54 = 2 × 3 × 3 × 3 = 2 × 3³
2. Prime factorization of 90:
- 90 = 2 × 3 × 3 × 5 = 2 × 3² × 5
#### (ii) Find the product of the common prime factors of these numbers:
1. Common prime factors of 54 and 90:
- Both have the factors 2 and 3.
- The highest power of common factor 3 is 3².
2. Product of common prime factors:
- Product = 2 × 3² = 2 × 9 = 18
#### (iii) Is the product the H.C.F. of the numbers?
1. HCF (Highest Common Factor) Calculation:
- The common factors identified above (2 and 3²) already confirm the HCF calculation directly.
2. Why is this the HCF?
- The HCF is the largest number that divides both 54 and 90 without a remainder.
- The prime factorizations of 54 and 90 share the factor 2 and 3².
3. As determined, the product of these common factors is indeed 18, so Yes, the product is the HCF of the numbers.
### 3. Distribution of 24 Snickers and 36 Cadburies
Bishwant distributes these equally among his friends. We need to find suitable numbers of friends.
#### (i) Possible numbers of friends to whom the snickers can be equally distributed:
1. Factors of 24:
- 1, 2, 3, 4, 6, 8, 12, 24
#### (ii) Possible numbers of friends to whom the cadburies can be equally distributed:
1. Factors of 36:
- 1, 2, 3, 4, 6, 9, 12, 18, 36
#### (iii) Possible numbers of friends to whom both the snickers and cadburies can be equally distributed:
1. Common factors between 24 and 36:
- Common factors = 1, 2, 3, 4, 6, 12
#### (iv) Greatest number of friends to whom the candies can be distributed equally:
1. Greatest common divisor (GCD) of 24 and 36:
- From the common factors, the greatest number is 12.
So, the greatest number of friends to whom the 24 snickers and 36 cadburies can be distributed equally is 12.