Properties of Whole Numbers
(iii) Among these common factors, which one is the greatest number
that divides 36 and 60 exactly?
b) 54 and 90 are two given numbers. Answer the following questions.
(i) Express each number as the product of their prime factors.
(ii) Find the product of the common prime factors of these numbers.
(iii) Is the product the H.C.F. of the numbers? Give reason.
a) On the occasion of Bishwant's birthday, he distributed 24 snickers and
36 cadburies equally to his friends.
(i) Write the possible numbers of friends to whom the snickers can be
equally distributed.
(ii) Write the possible numbers of friends to whom the cadburies can be
equally distributed.
(iii) What are the possible numbers of friends to whom the snickers and
cadburies can be equally distributed?
(iv) What is the greatest number of friends to whom the snickers and

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.