The people of Evenland never use odd digits. Instead of counting 1,2,3,4,5,6, an Evenlander counts 2, 4, 6, 8, 20, 22. What is an Evenlander's version of the integer 111?
(A) 822
(B) 828
(C) 840
(D) 842
(E) 824

Part C: Each correct answer is worth 8.
Solution 1

Suppose instead of using the digits $0,2,4,6,8$, that Evenlanders use the digits $0,1,2,3$, 4 (each of the previous digits divided by 2 ).

Then we can see that the Evenlanders' numbers system is our base 5 number system, with all of the digits doubled.

Writing 111 as a sum of powers of 5 , we see that $111=4\left(5^{2}\right)+2\left(5^{1}\right)+1\left(5^{0}\right)$, or 111 can be written as 421 in base 5 . Doubling the digits, the Evenlanders write 842 for the integer 111 .

## Solution 2

In order to determine the Evenlanders' version of 111, we need to find a pattern in the Evenlanders' numbers. So we write out the first several:

$$
2,4,6,8,20,22,24,26,28,40,42,44,46,48,60,62,64,66,68,80, \ldots
$$

which correspond to the integers 1 through 20 . We can see from these that the last digit of the Evenlanders' numbers has a 5 digit cycle $(2,4,6,8,0)$. Since we are looking for their version of 111, the last digit must be 2 .

In order to figure out the other digits, we need to extend our pattern. So continuing to count, we get

$80,82,84,86,88,200,202,204,206,208,220, \ldots$

for the integers 20 through 30 .

The numbers will then continue to have three digits with first digit 2 as the last two digits go from 00 to 88 . After 288 will come 400 and the cycle begins again.

How many three digit numbers beginning with 2 do the Evenlanders have? Since these are the numbers 200 through 288 , it is the same number as counting 00 through 88 , which is 25 numbers, since 88 corresponds to the integer 24 .

So 200 represents the integer 25 , which means that 400 represents the integer 50,600 will represent the integer 75, and 800 will represent the integer 100.

To get from the integer 100 to the integer 111, we can either count as Evenlanders from 800 upwards, or we can look for the 11th number in our original list, which is 42 (which represents the integer 11), and put an 8 at the front.

Therefore, the Evenlanders' version of 111 is 842 .

ANSWER: (D)