63: 9:: 574 : ?

Options

A. 73
B. 82
C. 84
D. 87

Explanation

To solve the analogy problem \( 63 : 9 :: 574 : ? \), we can follow these steps:

### Step 1: Understand the relationship in the first pair
The first pair is \( 63 : 9 \). We can express \( 63 \) in terms of \( 9 \):
\[ 63 = 7 \times 9 \]
This means that \( 63 \) is \( 7 \) times \( 9 \).

### Step 2: Set up the relationship for the second pair
Now, we need to find a similar relationship for \( 574 : ? \). We can express \( 574 \) in terms of a number \( x \):
\[ 574 = x \times ? \]
We want to find \( ? \) such that it maintains the same relationship as in the first pair.

### Step 3: Factor \( 574 \)
Next, we need to factor \( 574 \) to find a suitable \( x \):
\[ 574 = 2 \times 287 \]
We can further factor \( 287 \):
\[ 287 = 7 \times 41 \]
So, we can write:
\[ 574 = 2 \times 7 \times 41 \]

### Step 4: Identify a suitable \( x \)
From the factors, we see that \( 574 \) can be expressed as \( 82 \times 7 \) (since \( 82 = 2 \times 41 \)). Therefore, we can express \( 574 \) as:
\[ 574 = 82 \times 7 \]

### Step 5: Set up the final ratio
Now, we can write the second pair as:
\[ 574 : ? = 82 \times 7 : 82 \]
This means that \( ? \) should equal \( 7 \).

### Conclusion
Thus, the answer to the analogy \( 63 : 9 :: 574 : ? \) is:
\[ ? = 7 \]