Remainder when `3xx7^22+2xx10^22-44` is divisible by 18 is

Options

A. 16
B. 3
C. 12
D. 15

Answer

To find the remainder when \( 3 \times 7^{22} + 2 \times 10^{22} - 44 \) is divided by 18, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 3 \times 7^{22} + 2 \times 10^{22} - 44 \]

### Step 2: Substitute \( 7 \) and \( 10 \)
We can rewrite \( 7 \) and \( 10 \) in terms of their remainders when divided by 18:
\[ 7 \equiv 7 \mod 18 \quad \text{and} \quad 10 \equiv 10 \mod 18 \]
Thus, we can keep \( 7^{22} \) and \( 10^{22} \) as they are for now.

### Step 3: Calculate \( 7^{22} \mod 18 \)
To simplify \( 7^{22} \mod 18 \), we can use Euler's theorem. First, we find \( \phi(18) \):
\[ \phi(18) = 18 \left(1 - \frac{1}{2}\right) \left(1 - \frac{1}{3}\right) = 18 \times \frac{1}{2} \times \frac{2}{3} = 6 \]
Since \( 22 \mod 6 = 4 \), we have:
\[ 7^{22} \equiv 7^4 \mod 18 \]
Now, calculate \( 7^4 \):
\[ 7^2 = 49 \equiv 13 \mod 18 \]
\[ 7^4 = (7^2)^2 = 13^2 = 169 \equiv 7 \mod 18 \]
Thus,
\[ 7^{22} \equiv 7 \mod 18 \]

### Step 4: Calculate \( 10^{22} \mod 18 \)
Next, we calculate \( 10^{22} \mod 18 \):
Using Euler's theorem again, since \( 22 \mod 6 = 4 \):
\[ 10^{22} \equiv 10^4 \mod 18 \]
Now, calculate \( 10^4 \):
\[ 10^2 = 100 \equiv 10 \mod 18 \]
\[ 10^4 = (10^2)^2 = 10^2 \equiv 10 \mod 18 \]
Thus,
\[ 10^{22} \equiv 10 \mod 18 \]

### Step 5: Substitute back into the expression
Now substitute \( 7^{22} \) and \( 10^{22} \) back into the original expression:
\[ 3 \times 7^{22} + 2 \times 10^{22} - 44 \equiv 3 \times 7 + 2 \times 10 - 44 \mod 18 \]
Calculating this gives:
\[ 3 \times 7 = 21 \equiv 3 \mod 18 \]
\[ 2 \times 10 = 20 \equiv 2 \mod 18 \]
So,
\[ 3 + 2 - 44 \equiv 5 - 44 \mod 18 \]
Calculating \( -44 \mod 18 \):
\[ -44 \equiv -44 + 54 = 10 \mod 18 \]
Thus,
\[ 5 + 10 \equiv 15 \mod 18 \]

### Step 6: Final result
The final remainder when \( 3 \times 7^{22} + 2 \times 10^{22} - 44 \) is divided by 18 is:
\[ \boxed{15} \]