The number `334 xx 545 xx 7p` is divisible by 3340 if p is at least.

Options

A. 2
B. 4
C. 3
D. 1

Answer

To determine the least value of \( p \) such that the number \( 334 \times 545 \times 7p \) is divisible by \( 3340 \), we can follow these steps:

### Step 1: Factorize \( 3340 \)
First, we need to factorize \( 3340 \) to understand its prime factors.
\[ 3340 = 334 \times 10 \]
Now, we can factor \( 10 \) further:
\[ 10 = 2 \times 5 \]
Thus, we have:
\[ 3340 = 334 \times 2 \times 5 \]

### Step 2: Factorize \( 334 \)
Next, we factor \( 334 \):
\[ 334 = 2 \times 167 \]
So, we can write:
\[ 3340 = (2 \times 167) \times 2 \times 5 = 2^2 \times 5 \times 167 \]

### Step 3: Factorize \( 545 \)
Now, we factor \( 545 \):
\[ 545 = 5 \times 109 \]

### Step 4: Combine the Factors
Now we can combine the factors of \( 334 \), \( 545 \), and \( 7p \):
\[ 334 \times 545 \times 7p = (2 \times 167) \times (5 \times 109) \times (7p) \]
This gives us:
\[ = 2 \times 5 \times 7 \times 167 \times 109 \times p \]

### Step 5: Identify Required Factors for Divisibility
For \( 334 \times 545 \times 7p \) to be divisible by \( 3340 \), it must contain at least:
- \( 2^2 \) (from \( 3340 \))
- \( 5^1 \) (from \( 3340 \))
- \( 167^1 \) (from \( 3340 \))

### Step 6: Check Current Factors
From our product:
- We have \( 2^1 \) from \( 334 \).
- We have \( 5^1 \) from \( 545 \).
- We have \( 7 \) (not needed for divisibility).
- We have \( p \) which must provide the necessary factors.

### Step 7: Determine the Value of \( p \)
To satisfy the requirement of \( 2^2 \), we need one more factor of \( 2 \). Thus, we can set:
\[ p = 2 \]
This gives us:
\[ 7p = 7 \times 2 = 14 \]
Now we have:
- \( 2^2 \) from \( 334 \) and \( p \).
- \( 5^1 \) from \( 545 \).
- \( 167^1 \) from \( 334 \).

### Conclusion
Thus, the least value of \( p \) such that \( 334 \times 545 \times 7p \) is divisible by \( 3340 \) is:
\[ \boxed{2} \]