Find the greatest number that exactly divides 81, 144, and 162.

Options

A. 9
B. 27
C. 3
D. 81

Answer

To find the greatest number that exactly divides 81, 144, and 162, we need to determine the Highest Common Factor (HCF) of these three numbers. We can do this by following these steps:

### Step 1: Prime Factorization
We will start by finding the prime factorization of each number.

1. **Prime Factorization of 81:**
- 81 is divisible by 3.
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- So, the prime factorization of 81 is \( 3^4 \).

2. **Prime Factorization of 144:**
- 144 is divisible by 2.
- 144 ÷ 2 = 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 is divisible by 3.
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- So, the prime factorization of 144 is \( 2^4 \times 3^2 \).

3. **Prime Factorization of 162:**
- 162 is divisible by 2.
- 162 ÷ 2 = 81
- 81 is divisible by 3.
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- So, the prime factorization of 162 is \( 2^1 \times 3^4 \).

### Step 2: Identify Common Factors
Now, we will identify the common prime factors from the factorizations:

- **For 81:** \( 3^4 \)
- **For 144:** \( 2^4 \times 3^2 \)
- **For 162:** \( 2^1 \times 3^4 \)

The common prime factor is 3.

### Step 3: Determine the Lowest Power of Common Factors
Now, we take the lowest power of the common prime factors:

- For the prime factor 3, the lowest power is \( 3^2 \) (since 3 appears as \( 3^2 \) in 144).

### Step 4: Calculate the HCF
Now we calculate the HCF:

\[ \text{HCF} = 3^2 = 9 \]

Thus, the greatest number that exactly divides 81, 144, and 162 is **9**.

### Summary
The greatest number that exactly divides 81, 144, and 162 is **9**.

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