The Last Two Digits of the Number 9^9^9

To determine the last two digits of the number 9^9^9, we need to understand the properties of modular arithmetic and the concept of finding remainders.

Modular Arithmetic:
Modular arithmetic is a system of arithmetic for integers where numbers "wrap around" upon reaching a certain value, called the modulus. In this case, we are interested in finding the remainder when dividing a number by 100.

Finding Remainders:
To find the remainder when dividing a number by 100, we only need to consider the last two digits of the number. For example, the remainder of 123456 divided by 100 is 56.

Breaking Down the Exponent:
To tackle the problem of finding the last two digits of 9^9^9, we need to break down the exponent into smaller parts.

Step 1: Evaluating 9^9:
To find the last two digits of 9^9, we can use modular arithmetic. Let's find the remainder when 9^9 is divided by 100.

9^2 = 81
9^3 = 729
9^4 = (9^2)^2 = 81^2 = 6561

Continuing this pattern, we can calculate that 9^9 is equal to 387,420,489. Therefore, the remainder when dividing 9^9 by 100 is 89.

Step 2: Evaluating 9^9^9:
Now that we have found the remainder of 9^9, we can proceed to calculate the last two digits of 9^9^9.

Let's consider the remainder when 9^9^9 is divided by 100. We need to find the remainder of 89^9 when divided by 100.

To simplify the process, we can look for a pattern in the remainders when raising 89 to consecutive powers:

89^1 ≡ 89 (mod 100)
89^2 ≡ 21 (mod 100)
89^3 ≡ 69 (mod 100)
89^4 ≡ 41 (mod 100)
89^5 ≡ 49 (mod 100)
89^6 ≡ 61 (mod 100)
89^7 ≡ 89 (mod 100)

We can observe that the remainders repeat after every four powers. Therefore, we can determine the remainder of 89^9 by finding the remainder of 9 divided by 4, which is 1.

Hence, the remainder when calculating 9^9^9 is divided by 100 is the same as the remainder when 89^1 is divided by 100, which is 89.

The Last Two Digits:
Therefore, the last two digits of the number 9^9^9 are 89.