To encode a message, James first replaces each letter within the message with its corresponding number, where $A=1, B=2, \cdots, Y=25$, and $Z=26$. Next, James multiplies each number by 3 and then subtracts 5 , and continues this process a total of $n$ times. For example, when $n=2$ the letter $D$ is encoded to the number 16 .

If James encoded a four letter message to the four numbers 367205853 1339, what is the value of $n$ that he used?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
To encode a letter, James multiplies its corresponding number by 3 and then subtracts 5 , continuing this process a total of $n$ times.

To decode a number, the inverse operations must be performed in the opposite order.

The inverse operation of multiplication is division. The inverse operation of subtraction is addition.

Thus to decode a number, add 5 and then divide by 3 , and continue this process a total of $n$ times.

For example, when $n=1$ the number 4 (corresponding to the letter $D$ ) is encoded to $4 \times 3-5=7$. The number 7 is decoded by adding 5 and then dividing by 3 . We may check that this works by noting that $(7+5) \div 3=4$ (which corresponds to the letter $D$ ), as required.

Each letter of James' original message corresponds to a number from 1 to 26 , inclusive.

To determine the value of $n$, we may begin with the four given encoded numbers $(367,205,853,1339)$ and continue to apply the decoding process until the resulting numbers are each equal to a number from 1 to 26, inclusive (since each letter of the original message corresponds to a number from 1 to 26 , inclusive).

We show this work in the table below.

|  | 367 | 205 | 853 | 1339 |
| :---: | :---: | :---: | :---: | :---: |
| $n=1$ | $(367+5) \div 3=124$ | $(205+5) \div 3=70$ | $(853+5) \div 3=286$ | $(1339+5) \div 3=448$ |
| $n=2$ | $(124+5) \div 3=43$ | $(70+5) \div 3=25$ | $(286+5) \div 3=97$ | $(448+5) \div 3=151$ |
| $n=3$ | 16 | 10 | 34 | 52 |
| $n=4$ | 7 | 5 | 13 | 19 |
| $n=5$ | 4 | $\frac{10}{3}$ | 6 | 8 |

From the table, the first value of $n$ for which each of the four numbers is from 1 to 26 inclusive, is $n=4$.

Further, we note that if $n=5$ the original number corresponding to 205 is $\frac{10}{3}$, which is not possible.

Therefore, the value of $n$ used by James was 4 .

(Although the question did not ask for the original message, the letters corresponding to $7,5,13,19$ are $G, E, M, S$.)

ANswer: $(\mathrm{C})$