For every question, four probable answers as (A), (B), (C) and (D) are given. Only one out of these is correct. Choose the correct answer.
The sum of H.C.F. and L.C.M of 45, 60 and 75 is:

Options

A. 330
B. 960
C. 915
D. 630

Answer

To find the sum of the H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) of the numbers 45, 60, and 75, we will follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

- **For 45:**
- 45 = 3 × 15
- 15 = 3 × 5
- Therefore, the prime factorization of 45 is \( 3^2 \times 5^1 \).

- **For 60:**
- 60 = 2 × 30
- 30 = 2 × 15
- 15 = 3 × 5
- Therefore, the prime factorization of 60 is \( 2^2 \times 3^1 \times 5^1 \).

- **For 75:**
- 75 = 3 × 25
- 25 = 5 × 5
- Therefore, the prime factorization of 75 is \( 3^1 \times 5^2 \).

### Step 2: Finding H.C.F.
To find the H.C.F., we take the lowest power of each common prime factor.

- Common prime factors: 3 and 5
- For 3: the lowest power is \( 3^1 \) (from 75).
- For 5: the lowest power is \( 5^1 \) (from both 45 and 60).

Thus, the H.C.F. = \( 3^1 \times 5^1 = 3 \times 5 = 15 \).

### Step 3: Finding L.C.M.
To find the L.C.M., we take the highest power of each prime factor.

- For 2: the highest power is \( 2^2 \) (from 60).
- For 3: the highest power is \( 3^2 \) (from 45).
- For 5: the highest power is \( 5^2 \) (from 75).

Thus, the L.C.M. = \( 2^2 \times 3^2 \times 5^2 \).

Calculating this:
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 5^2 = 25 \)

Now, \( L.C.M. = 4 \times 9 \times 25 \).

Calculating step-by-step:
1. \( 4 \times 9 = 36 \)
2. \( 36 \times 25 = 900 \)

So, L.C.M. = 900.

### Step 4: Sum of H.C.F. and L.C.M.
Now, we sum the H.C.F. and L.C.M.:
- H.C.F. = 15
- L.C.M. = 900

Sum = \( 15 + 900 = 915 \).

### Final Answer
The sum of H.C.F. and L.C.M. of 45, 60, and 75 is **915**.