Compare the following quantities: **Quantity I:** Unit digit of the number $1333^{2755}$ **Quantity II:** Unit digit of the number $127^{4352}$ Which of the following is true? (a) Quantity I Quantity II (b) Quantity I

Question

Compare the following quantities:

**Quantity I:** Unit digit of the number $1333^{2755}$

**Quantity II:** Unit digit of the number $127^{4352}$

Which of the following is true?

(a) Quantity I > Quantity II

(b) Quantity I < Quantity II

(c) Quantity I ≥ Quantity II

(d) Quantity I ≤ Quantity II

(e) Quantity I = Quantity II or No relation

Accepted Answer

**Solution:**

1. **Quantity I:** Find the unit digit of $1333^{2755}$
   - The unit digit of the base $1333$ is $3$.
   - The pattern of unit digits for powers of $3$ repeats every 4 powers:
     - $3^1 = 3$ (unit digit 3)
     - $3^2 = 9$ (unit digit 9)
     - $3^3 = 27$ (unit digit 7)
     - $3^4 = 81$ (unit digit 1)
     - Then it repeats: 3, 9, 7, 1, ...

2. Find the remainder when the exponent $2755$ is divided by 4:
   $$2755 mod 4 = 3$$

3. The unit digit corresponds to the 3rd number in the cycle, which is $7$.

4. **Quantity II:** Find the unit digit of $127^{4352}$
   - The unit digit of the base $127$ is $7$.
   - The pattern of unit digits for powers of $7$ repeats every 4 powers:
     - $7^1 = 7$ (unit digit 7)
     - $7^2 = 49$ (unit digit 9)
     - $7^3 = 343$ (unit digit 3)
     - $7^4 = 2401$ (unit digit 1)
     - Then it repeats: 7, 9, 3, 1, ...

5. Find the remainder when the exponent $4352$ is divided by 4:
   $$4352 mod 4 = 0$$

6. When remainder is 0, the unit digit corresponds to the 4th number in the cycle, which is $1$.

7. Comparing the unit digits:
   - Quantity I unit digit = $7$
   - Quantity II unit digit = $1$

Since $7 > 1$, **Quantity I > Quantity II**.

**Answer:** (a) Quantity I > Quantity II

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