Find the remainder when `7^(52)` is divided by 2402.

Answer

To find the remainder when \( 7^{52} \) is divided by \( 2402 \), we can follow these steps:

### Step 1: Factor 2402
First, we need to factor \( 2402 \):
\[ 2402 = 2 \times 1201 \]
Next, we check if \( 1201 \) is a prime number. It turns out that \( 1201 \) is indeed a prime number.

### Step 2: Use the Binomial Theorem
We can express \( 7^{52} \) in a form that allows us to use the Binomial Theorem. We can write \( 7^{52} \) as:
\[ 7^{52} = (2402 - 1)^{13} \]
This is because \( 7^4 = 2401 \) and \( 52 = 4 \times 13 \).

### Step 3: Expand using the Binomial Theorem
Using the Binomial Theorem, we expand \( (2402 - 1)^{13} \):
\[ (2402 - 1)^{13} = \sum_{k=0}^{13} \binom{13}{k} (2402)^{13-k} (-1)^k \]
When we divide this expansion by \( 2402 \), all terms where \( k < 13 \) will be divisible by \( 2402 \) and will contribute \( 0 \) to the remainder. The only term that does not contribute is when \( k = 13 \):
\[ (-1)^{13} = -1 \]

### Step 4: Determine the Remainder
Thus, the remainder when \( 7^{52} \) is divided by \( 2402 \) is:
\[ -1 \]
However, since remainders are typically expressed as non-negative integers, we can convert \( -1 \) to a positive equivalent by adding \( 2402 \):
\[ Remainder = 2402 - 1 = 2401 \]

### Final Answer
The remainder when \( 7^{52} \) is divided by \( 2402 \) is \( 2401 \).

---