To find the H.C.F. (Highest Common Factor) of the numbers \( x \), \( y \), and \( z \), we will follow these steps:
### Step 1: Write down the prime factorization of each number.
- \( x = 2^3 \times 3^2 \times 5^3 \times 7^3 \)
- \( y = 2^2 \times 3^3 \times 5^4 \times 7^3 \)
- \( z = 2^4 \times 3^4 \times 5^2 \times 7^5 \)
### Step 2: Identify the minimum power of each prime factor.
To find the H.C.F., we take the minimum power of each prime factor from the factorizations of \( x \), \( y \), and \( z \):
- For \( 2 \): The powers are \( 3, 2, 4 \). The minimum is \( 2 \).
- For \( 3 \): The powers are \( 2, 3, 4 \). The minimum is \( 2 \).
- For \( 5 \): The powers are \( 3, 4, 2 \). The minimum is \( 2 \).
- For \( 7 \): The powers are \( 3, 3, 5 \). The minimum is \( 3 \).
### Step 3: Write down the H.C.F. using the minimum powers.
Now we can write the H.C.F. using the minimum powers we found:
\[
\text{H.C.F.} = 2^2 \times 3^2 \times 5^2 \times 7^3
\]
### Step 4: Calculate the numerical value of the H.C.F.
Now we will calculate the numerical value of the H.C.F.:
- Calculate \( 2^2 = 4 \)
- Calculate \( 3^2 = 9 \)
- Calculate \( 5^2 = 25 \)
- Calculate \( 7^3 = 343 \)
Now multiply these values together:
\[
\text{H.C.F.} = 4 \times 9 \times 25 \times 343
\]
### Step 5: Perform the multiplication step by step.
1. First, calculate \( 4 \times 9 = 36 \).
2. Next, calculate \( 36 \times 25 = 900 \).
3. Finally, calculate \( 900 \times 343 \).
To simplify \( 900 \times 343 \):
- \( 900 = 9 \times 100 = 9 \times 10^2 \)
- \( 343 = 7^3 \)
So, \( 900 \times 343 = 900 \times 343 = 308700 \).
Thus, the H.C.F. of \( x \), \( y \), and \( z \) is:
\[
\text{H.C.F.} = 900 \times 343 = 308700
\]
### Final Answer:
The H.C.F. of \( x \), \( y \), and \( z \) is \( 308700 \).
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