What would be the remainder when `10^(6)-12` is divided by 9 ?

Options

A. 5
B. 4
C. 7
D. 3

Answer

To find the remainder when \( 10^6 - 12 \) is divided by 9, we can follow these steps:

### Step 1: Calculate \( 10^6 \mod 9 \)
First, we need to find the remainder of \( 10^6 \) when divided by 9. We can simplify \( 10 \mod 9 \):
\[ 10 \mod 9 = 1 \]
Thus, we can rewrite \( 10^6 \) as:
\[ 10^6 \mod 9 = (10 \mod 9)^6 = 1^6 = 1 \]

### Step 2: Calculate \( 12 \mod 9 \)
Next, we find the remainder of 12 when divided by 9:
\[ 12 \mod 9 = 3 \]

### Step 3: Combine the results
Now, we substitute the results from steps 1 and 2 into the expression \( 10^6 - 12 \):
\[ 10^6 - 12 \mod 9 = (1 - 3) \mod 9 \]
Calculating this gives:
\[ 1 - 3 = -2 \]

### Step 4: Adjust for negative remainder
Since remainders cannot be negative, we need to convert \(-2\) into a positive remainder by adding 9:
\[ -2 + 9 = 7 \]

### Conclusion
Thus, the remainder when \( 10^6 - 12 \) is divided by 9 is:
\[ \boxed{7} \]