The HCF of `8^(7),8^(-63),8^(-60),8^(-3)` is

Options

A. A)`8^(-6)`
B. B)`8^(-3)`
C. C)`8^(-7)`
D. D)`8^(-63)`

Answer

To find the HCF (Highest Common Factor) of the numbers \( 8^7, 8^{-63}, 8^{-60}, 8^{-3} \), we can follow these steps:

### Step 1: Write the numbers in exponential form
The numbers are already in exponential form:
- \( 8^7 \)
- \( 8^{-63} \)
- \( 8^{-60} \)
- \( 8^{-3} \)

### Step 2: Identify the exponents
The exponents of the numbers are:
- For \( 8^7 \), the exponent is \( 7 \).
- For \( 8^{-63} \), the exponent is \( -63 \).
- For \( 8^{-60} \), the exponent is \( -60 \).
- For \( 8^{-3} \), the exponent is \( -3 \).

### Step 3: Find the HCF of the exponents
To find the HCF of the numbers, we need to find the HCF of the exponents \( 7, -63, -60, -3 \).

1. **Convert all exponents to positive values** (since HCF is typically considered with positive integers):
- The positive equivalents are \( 7, 63, 60, 3 \).

2. **Find the HCF of these numbers**:
- The factors of \( 7 \) are \( 1, 7 \).
- The factors of \( 63 \) are \( 1, 3, 7, 9, 21, 63 \).
- The factors of \( 60 \) are \( 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 \).
- The factors of \( 3 \) are \( 1, 3 \).

The common factors among these numbers are \( 1 \) and \( 3 \). The highest of these is \( 3 \).

### Step 4: Write the HCF in terms of the base
Since the HCF of the exponents is \( 3 \), we can express the HCF of the original numbers as:
\[ HCF = 8^{3} \]

### Final Answer
Thus, the HCF of \( 8^7, 8^{-63}, 8^{-60}, 8^{-3} \) is:
\[ 8^{3} \]