What is the **unit digit** in the expansion of $67^{32}$?

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**Solution:**

1. The unit digit of a number raised to a power depends only on the unit digit of the base.

2. The base is $67$, so the unit digit is $7$.

3. We need to find the unit digit of $7^{32}$.

4. 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)

5. After every 4 powers, the pattern repeats: 7, 9, 3, 1

6. Find the remainder when $32$ is divided by $4$:
   $$32 \div 4 = 8 	ext{ remainder } 0$$

7. When the remainder is 0, the unit digit corresponds to the 4th element in the pattern, which is $1$.

**Answer:** $1$

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