NO CALCULATOR!

1) Find the last two digits of the number 11^2016 .

To find the last two digits of the number 11^2016 without using a calculator, we can use modular arithmetic.

First, let's find a pattern by calculating the powers of 11 modulo 100 (the two-digit number we are interested in).

11^1 ≡ 11 (mod 100)
11^2 ≡ 21 (mod 100)
11^3 ≡ 31 (mod 100)
11^4 ≡ 41 (mod 100)
11^5 ≡ 51 (mod 100)
...
11^10 ≡ 91 (mod 100)
11^11 ≡ 01 (mod 100)
11^12 ≡ 11 (mod 100)

You might notice that after every 10 powers, the last two digits repeat. This pattern is due to the fact that 11^10 ≡ 1 (mod 100) by Euler's theorem.

Now, we need to find the remainder when dividing 2016 by 10, which is 6. This tells us that we need to find the value of 11^6 (mod 100) to get the last two digits of 11^2016.

Using the pattern we found earlier:
11^6 ≡ 21 (mod 100)

Therefore, we can conclude that the last two digits of 11^2016 are "21" without using a calculator.