The remainder when `27^(10)+7^(51)` is divided by `10`

Options

A. `4`
B. `6`
C. `9`
D. `2`

Explanation

To find the remainder when \( 27^{10} + 7^{51} \) is divided by \( 10 \), we can simplify each term separately and then combine the results.

### Step 1: Simplifying \( 27^{10} \mod 10 \)

First, we note that:
\[ 27 \equiv 7 \mod 10 \]
Thus, we can replace \( 27 \) with \( 7 \):
\[ 27^{10} \equiv 7^{10} \mod 10 \]

### Step 2: Finding \( 7^{10} \mod 10 \)

Next, we can find \( 7^{10} \mod 10 \) using the pattern of the powers of \( 7 \) modulo \( 10 \):
- \( 7^1 \equiv 7 \mod 10 \)
- \( 7^2 \equiv 49 \equiv 9 \mod 10 \)
- \( 7^3 \equiv 7 \times 9 \equiv 63 \equiv 3 \mod 10 \)
- \( 7^4 \equiv 7 \times 3 \equiv 21 \equiv 1 \mod 10 \)

We see that \( 7^4 \equiv 1 \mod 10 \). Therefore, the powers of \( 7 \) repeat every \( 4 \):
\[ 7^{10} = 7^{4 \times 2 + 2} = (7^4)^2 \times 7^2 \equiv 1^2 \times 9 \equiv 9 \mod 10 \]

### Step 3: Simplifying \( 7^{51} \mod 10 \)

Now, we calculate \( 7^{51} \mod 10 \):
\[ 51 \mod 4 = 3 \]
Thus,
\[ 7^{51} \equiv 7^3 \mod 10 \]
From our previous calculations:
\[ 7^3 \equiv 3 \mod 10 \]

### Step 4: Combining the results

Now we can combine the results:
\[ 27^{10} + 7^{51} \equiv 7^{10} + 7^{51} \equiv 9 + 3 \mod 10 \]
\[ 9 + 3 = 12 \]

### Step 5: Finding the final remainder

Finally, we find the remainder when \( 12 \) is divided by \( 10 \):
\[ 12 \mod 10 = 2 \]

Thus, the remainder when \( 27^{10} + 7^{51} \) is divided by \( 10 \) is \( \boxed{2} \).

---