1. **Problem Statement:** Find the digit $D$ such that the last digit of $(54D)^{100}$ is 1. 2. **Understanding the problem:** The last digit of a number raised to a power depends only on the last digit of the base number. Here, the base number ends with digit $D$. 3. **Key formula:** The last digit of $a^n$ depends on the last digit of $a$ and the pattern of its powers modulo 10. 4. **Step 1: Express the base number's last digit:** The base number ends with digit $D$, so the last digit of the base is $D$. 5. **Step 2: Analyze the last digit of $(54D)^{100}$:** Since the last digit of $54D$ is $D$, the last digit of $(54D)^{100}$ is the same as the last digit of $D^{100}$. 6. **Step 3: Find $D$ such that $D^{100}$ ends with 1:** We want $D^{100} \equiv 1 \pmod{10}$. 7. **Step 4: Check possible values of $D$ (0 to 9) for $D^{100} \equiv 1 \pmod{10}$:** - $0^{100} = 0$ (ends with 0) - $1^{100} = 1$ (ends with 1) - $2^{100}$ ends with 6 (since powers of 2 cycle every 4: 2,4,8,6) - $3^{100}$ ends with 1 (powers of 3 cycle every 4: 3,9,7,1) - $4^{100}$ ends with 6 (cycle 4,6) - $5^{100}$ ends with 5 - $6^{100}$ ends with 6 - $7^{100}$ ends with 1 (cycle 7,9,3,1) - $8^{100}$ ends with 6 (cycle 8,4,2,6) - $9^{100}$ ends with 1 (cycle 9,1) So possible $D$ values are 1, 3, 7, or 9. 8. **Step 5: Use Statement I: $D > 5$** - Possible $D$ values from step 7 that satisfy $D > 5$ are 7 and 9. - So Statement I narrows $D$ to 7 or 9 but does not give a unique value. 9. **Step 6: Use Statement II: $D$ is a multiple of 3** - Possible $D$ values from step 7 that are multiples of 3 are 3 only (since 9 is multiple of 3 but 9 ends with 1, so 9 is possible too). - From step 7, $D$ can be 3 or 9. - So Statement II narrows $D$ to 3 or 9 but does not give a unique value. 10. **Step 7: Use both Statements together:** - $D > 5$ and $D$ multiple of 3 means $D$ can only be 9. 11. **Step 8: Conclusion:** - Neither Statement I nor Statement II alone is sufficient to find unique $D$. - Both Statements together uniquely determine $D=9$. **Final answer:** The correct choice is (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.