If a=` ( 2^(2) xx 3^(3) xx 5^(4)) and b = (2^(3) xx 3^(2) xx 5) ` then HCF `(a, b)` ?

Options

A. 90
B. 180
C. 360
D. 540

Explanation

To find the HCF (Highest Common Factor) of the numbers \( a \) and \( b \) given in the question, we will follow these steps:

### Step 1: Write down the prime factorization of \( a \) and \( b \).

Given:
- \( a = 2^2 \times 3^3 \times 5^4 \)
- \( b = 2^3 \times 3^2 \times 5^1 \)

### Step 2: Identify the common prime factors.

The common prime factors between \( a \) and \( b \) are:
- For the prime factor \( 2 \): The powers are \( 2 \) in \( a \) and \( 3 \) in \( b \).
- For the prime factor \( 3 \): The powers are \( 3 \) in \( a \) and \( 2 \) in \( b \).
- For the prime factor \( 5 \): The powers are \( 4 \) in \( a \) and \( 1 \) in \( b \).

### Step 3: Choose the lowest power for each common prime factor.

- For \( 2 \): The minimum power is \( \min(2, 3) = 2 \).
- For \( 3 \): The minimum power is \( \min(3, 2) = 2 \).
- For \( 5 \): The minimum power is \( \min(4, 1) = 1 \).

### Step 4: Write the HCF using the lowest powers.

Now, we can write the HCF as:
\[ \text{HCF}(a, b) = 2^{\min(2, 3)} \times 3^{\min(3, 2)} \times 5^{\min(4, 1)} = 2^2 \times 3^2 \times 5^1 \]

### Step 5: Calculate the numerical value of the HCF.

Now, we calculate:
\[ 2^2 = 4, \quad 3^2 = 9, \quad 5^1 = 5 \]
Now, multiply these together:
\[ \text{HCF}(a, b) = 4 \times 9 \times 5 \]

Calculating step-by-step:
1. \( 4 \times 9 = 36 \)
2. \( 36 \times 5 = 180 \)

Thus, the HCF of \( a \) and \( b \) is \( 180 \).

### Final Answer:
\[ \text{HCF}(a, b) = 180 \]

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