The problem presents a subtraction problem in base 7, where one digit in the result is missing. We are given the subtraction $5162_7 - 2644_7 = 2*15_7$, and we need to find the value of the missing digit represented by the asterisk.

1. Problem Description

The problem presents a subtraction problem in base 7, where one digit in the result is missing. We are given the subtraction $5162_7 - 2644_7 = 2*15_7$, and we need to find the value of the missing digit represented by the asterisk.

2. Solution Steps

We will perform the subtraction in base

7. First, we look at the rightmost column: $2 - 4$. Since $2 < 4$, we need to borrow from the next column. So, we borrow 1 from the 6, which becomes

5. The 2 becomes $2 + 7 = 9$. Then $9 - 4 = 5$. So the rightmost digit of the result is indeed

5. Next, we look at the second column from the right. We now have $5 - 4 = 1$, which matches the result.

Now, we look at the third column from the right. We have $1 - 6$. Since $1 < 6$, we need to borrow from the next column. We borrow 1 from the 5, which becomes

4. The 1 becomes $1 + 7 = 8$. Then $8 - 6 = 2$. So the missing digit is

2. Finally, we look at the leftmost column. We have $4 - 2 = 2$, which matches the result.

We can write it out as:

$5162_7 - 2644_7$

$= 5 \cdot 7^3 + 1 \cdot 7^2 + 6 \cdot 7^1 + 2 \cdot 7^0 - (2 \cdot 7^3 + 6 \cdot 7^2 + 4 \cdot 7^1 + 4 \cdot 7^0)$

$= (5-2) \cdot 7^3 + (1-6) \cdot 7^2 + (6-4) \cdot 7^1 + (2-4) \cdot 7^0$

$= 3 \cdot 7^3 - 5 \cdot 7^2 + 2 \cdot 7^1 - 2 \cdot 7^0$

Borrow 1 from $7^3$, $3 \cdot 7^3 = 2 \cdot 7^3 + 7 \cdot 7^2$

$= 2 \cdot 7^3 + (7-5) \cdot 7^2 + 2 \cdot 7^1 - 2 \cdot 7^0$

$= 2 \cdot 7^3 + 2 \cdot 7^2 + 2 \cdot 7^1 - 2 \cdot 7^0$

Borrow 1 from $7^1$, $2 \cdot 7^1 = 1 \cdot 7^1 + 7 \cdot 7^0$

$= 2 \cdot 7^3 + 2 \cdot 7^2 + 1 \cdot 7^1 + (7-2) \cdot 7^0$

$= 2 \cdot 7^3 + 2 \cdot 7^2 + 1 \cdot 7^1 + 5 \cdot 7^0$

$= 2215_7$

3. Final Answer

C. 2