Decoding the FACTOR Code

The problem asks us to find a pattern used to encode the word 'FACTOR' into the number '186151203' and then apply this pattern to encode the word 'SQUARE'. The first step is to determine the numerical value of each letter based on its position in the alphabet (A=1, B=2, ..., Z=26).

For the word FACTOR:

• F = 6
• A = 1
• C = 3
• T = 20
• O = 15
• R = 18

The resulting number is 186151203. By comparing the letter positions {6, 1, 3, 20, 15, 18} with the code, we can deduce that the code is formed by concatenating the positions of the letters in a specific order. The order that successfully matches the code is R, F, O, A, T, C.

Let's verify this order:

• Position of R = 18
• Position of F = 6
• Position of O = 15
• Position of A = 1
• Position of T = 20
• Position of C = 3

Concatenating these numbers in the sequence R, F, O, A, T, C yields 18 followed by 6, then 15, then 1, then 20, then 3:

186151203 = 186151203.

This process can be represented using concatenation notation: $18 \oplus 6 \oplus 15 \oplus 1 \oplus 20 \oplus 3 = 186151203$.

Applying the Pattern to SQUARE

Now, we apply the identified logic to the word 'SQUARE'. First, list the alphabetical positions of its letters:

• S = 19
• Q = 17
• U = 21
• A = 1
• R = 18
• E = 5

The set of positions for the letters in SQUARE is {19, 17, 21, 1, 18, 5}.

To find the correct coded number for SQUARE, we determine the order in which these positions are concatenated. Based on the provided options, the number 5191817121 is the correct code. We can deduce that this number is formed by concatenating the positions in the order E, S, R, Q, A, U.

Let's verify this breakdown:

• Position of E = 5
• Position of S = 19
• Position of R = 18
• Position of Q = 17
• Position of A = 1
• Position of U = 21

Concatenating these values in this specific order (E, S, R, Q, A, U) results in:

5191817121 = 5191817121.

Using concatenation notation: $5 \oplus 19 \oplus 18 \oplus 17 \oplus 1 \oplus 21 = 5191817121$.

Solution Summary

The code language works by assigning each letter its standard alphabetical position (A=1, Z=26) and then concatenating these numbers. The sequence of concatenation for the letters is specific and needs to be determined by analyzing the example provided. For the word 'SQUARE', applying this logic results in the number 5191817121.