Remainder when 17^200 is divided by 18:

To find the remainder when 17^200 is divided by 18, we need to understand the concept of modular arithmetic. Modular arithmetic is a system of arithmetic in which numbers "wrap around" after reaching a certain value, called the modulus. In this case, the modulus is 18.

To solve this problem, we can use the concept of modular exponentiation. Modular exponentiation allows us to calculate the remainder when a number is raised to a power and divided by a modulus.

Step 1: Understanding the Problem
We are given the expression 17^200 and we need to find the remainder when it is divided by 18.

Step 2: Applying Modular Exponentiation
To calculate the remainder when 17^200 is divided by 18, we can use the following formula:

(a^b) mod n = ((a mod n)^b) mod n

In this formula, a is the base, b is the exponent, and n is the modulus.

Step 3: Simplifying the Expression
Applying the formula, we have:

(17^200) mod 18 = ((17 mod 18)^200) mod 18

Now let's simplify the expression further.

Step 4: Simplifying the Base
To find the remainder when 17 is divided by 18, we can simply divide 17 by 18:

17 mod 18 = 17

Therefore, we can rewrite the expression as:

(17^200) mod 18 = (17^200) mod 18

Step 5: Calculating the Remainder
To calculate the remainder when 17^200 is divided by 18, we can use a method called repeated squaring. Repeated squaring allows us to calculate large exponents efficiently.

We start by calculating the remainder when 17^1 is divided by 18:

(17^1) mod 18 = 17

Then, we square this remainder:

(17^2) mod 18 = (17 * 17) mod 18 = 289 mod 18 = 1

Next, we square the result again:

(17^4) mod 18 = (1 * 1) mod 18 = 1

We continue this process until we reach the desired exponent of 200:

(17^8) mod 18 = 1
(17^16) mod 18 = 1
(17^32) mod 18 = 1
...

We observe a pattern here. The remainder remains 1 when we keep squaring the result. Therefore:

(17^200) mod 18 = (17^32 * 17^32 * 17^32 * 17^32 * 17^32 * 17^32) mod 18 = (1 * 1 * 1 * 1 * 1 * 1) mod 18 = 1

Step 6: Final Answer
The remainder when 17^200 is divided by 18 is 1.

To summarize:
- We used the concept of modular exponentiation to find the remainder when 17^200 is divided by 18.
- By repeatedly squaring the base and reducing it modulo 18, we found that the remainder