Correct Option (B)

Let the symbol ⊗ represent a single digit, denoted by 'x'. The given sum can be translated into an algebraic equation by considering the place value of each term:

⊗ = x
1 ⊗ = 10 × 1 + x = 10 + x
5 ⊗ = 10 × 5 + x = 50 + x
⊗ ⊗ = 10 × x + x = 11x
⊗ 1 = 10 × x + 1 = 10x + 1

The sum equals 1 ⊗ ⊗, which translates to:

1 ⊗ ⊗ = 100 × 1 + 10 × x + x = 100 + 11x

Now, set up the equation:

x + (10 + x) + (50 + x) + (11x) + (10x + 1) = 100 + 11x

Combine like terms on the left-hand side (LHS):

(x + x + x + 11x + 10x) + (10 + 50 + 1) = 100 + 11x

24x + 61 = 100 + 11x

Subtract 11x from both sides:

24x - 11x + 61 = 100

13x + 61 = 100

Subtract 61 from both sides:

13x = 100 - 61

13x = 39

Divide by 13:

x = 39 / 13

x = 3

Thus, the symbol ⊗ stands for the digit 3. Substituting 3 into the original sum yields: 3 + 13 + 53 + 33 + 31 = 133, which matches 1 ⊗ ⊗ (133).

Incorrect Options:

Substituting the values from the other options into the equation does not satisfy the equality:

Option A: 2
If ⊗ = 2, the sum is 2 + 12 + 52 + 22 + 21 = 109. The right-hand side (1 ⊗ ⊗) would be 122. Since 109 ≠ 122, 2 is incorrect.
Option C: 4
If ⊗ = 4, the sum is 4 + 14 + 54 + 44 + 41 = 157. The right-hand side (1 ⊗ ⊗) would be 144. Since 157 ≠ 144, 4 is incorrect.
Option D: 5
If ⊗ = 5, the sum is 5 + 15 + 55 + 55 + 51 = 181. The right-hand side (1 ⊗ ⊗) would be 155. Since 181 ≠ 155, 5 is incorrect.