If `(67^67 + 67)` is divided by 68, the re- mainder is:

Options

A. 1
B. 67
C. 63
D. 66

Answer

To find the remainder when \(67^{67} + 67\) is divided by \(68\), we can follow these steps:

### Step 1: Simplify \(67\) modulo \(68\)
Since \(67\) is just \(1\) less than \(68\), we can express it as:
\[ 67 \equiv -1 \mod{68} \]

### Step 2: Substitute \(67\) in the expression
Now, we substitute \(67\) in the expression \(67^{67} + 67\):
\[ 67^{67} + 67 \equiv (-1)^{67} + (-1) \mod{68} \]

### Step 3: Calculate \((-1)^{67}\)
Since \(67\) is an odd number, we have:
\[ (-1)^{67} = -1 \]

### Step 4: Substitute back into the expression
Now, substituting back, we get:
\[ (-1) + (-1) = -2 \]

### Step 5: Find the remainder when divided by \(68\)
Now we need to find the remainder of \(-2\) when divided by \(68\). Since remainders must be non-negative, we can convert \(-2\) to a positive remainder by adding \(68\):
\[ -2 + 68 = 66 \]

Thus, the remainder when \(67^{67} + 67\) is divided by \(68\) is:
\[ \boxed{66} \]