Let's evaluate each of the statements separately to determine their validity.

Statement I: (D6C1)₁₆ = (54977)₁₀

To verify this statement, we need to convert the hexadecimal number (D6C1)₁₆ to its decimal equivalent.

A hexadecimal (base 16) number has digits 0-9 and A-F, where A=10, B=11, ..., F=15 in decimal.

The conversion from hexadecimal to decimal is done using the formula:

\(N = \sum_{i=0}^{n-1} a_i \times 16^i\), where \(a_i\) is the digit at index \(i\)

1. The rightmost digit (1) is at \(16^0\): \(1 \times 16^0 = 1\)
2. The next digit (C) is at \(16^1\): \(12 \times 16^1 = 192\) (since C=12 in decimal)
3. The next digit (6) is at \(16^2\): \(6 \times 16^2 = 1536\)
4. The leftmost digit (D) is at \(16^3\): \(13 \times 16^3 = 53248\) (since D=13 in decimal)

Summing these values gives:

\(53248 + 1536 + 192 + 1 = 54977\)

Thus, Statement I is true.

Statement II: (41819)₁₀ = (A35B)₁₆

To verify this statement, we need to convert the decimal number (41819)₁₀ to hexadecimal.

The method involves dividing the number by 16 and keeping track of the remainders.

1. 41819 divided by 16 equals 2613 with a remainder of 11. The remainder 11 corresponds to B in hex.
2. 2613 divided by 16 equals 163 with a remainder of 5.
3. 163 divided by 16 equals 10 with a remainder of 3.
4. 10 divided by 16 equals 0 with a remainder of 10. The remainder 10 corresponds to A in hex.

Reading the remainders from last to first gives the hexadecimal equivalent: (A35B)₁₆.

Thus, Statement II is also true.

Conclusion

Both Statement I and Statement II are true.