In each of the following numbers, replace ` **` by the smallest number to make it divisible by 3 :
(i) ` 27 **4 ` , (ii) `53** 46` , (iii) ` 8 **711` , (iv) `62 **35` , (v) `234**17` , (vi) `6**1054`

Answer

To solve the problem of replacing `**` with the smallest number to make each given number divisible by 3, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Rule of Divisibility by 3**:
A number is divisible by 3 if the sum of its digits is divisible by 3.

2. **For each number, calculate the sum of the known digits and include `**` as a variable. Then, determine the smallest digit (0-9) that can replace `**` to make the total sum divisible by 3.**

---

**(i) For `27**4`:**
- Known digits: 2, 7, 4
- Sum = 2 + 7 + 4 + `**` = 13 + `**`
- To find the smallest number to replace `**`, we need to find the smallest digit that makes (13 + `**`) divisible by 3.
- Testing values:
- If `**` = 0 → 13 + 0 = 13 (not divisible by 3)
- If `**` = 1 → 13 + 1 = 14 (not divisible by 3)
- If `**` = 2 → 13 + 2 = 15 (divisible by 3)
- **Replace `**` with 2.**

---

**(ii) For `53**46`:**
- Known digits: 5, 3, 4, 6
- Sum = 5 + 3 + 4 + 6 + `**` = 18 + `**`
- Since 18 is already divisible by 3, we can replace `**` with 0.
- **Replace `**` with 0.**

---

**(iii) For `8**711`:**
- Known digits: 8, 7, 1, 1
- Sum = 8 + 7 + 1 + 1 + `**` = 17 + `**`
- Testing values:
- If `**` = 0 → 17 + 0 = 17 (not divisible by 3)
- If `**` = 1 → 17 + 1 = 18 (divisible by 3)
- **Replace `**` with 1.**

---

**(iv) For `62**35`:**
- Known digits: 6, 2, 3, 5
- Sum = 6 + 2 + 3 + 5 + `**` = 16 + `**`
- Testing values:
- If `**` = 0 → 16 + 0 = 16 (not divisible by 3)
- If `**` = 1 → 16 + 1 = 17 (not divisible by 3)
- If `**` = 2 → 16 + 2 = 18 (divisible by 3)
- **Replace `**` with 2.**

---

**(v) For `234**17`:**
- Known digits: 2, 3, 4, 1, 7
- Sum = 2 + 3 + 4 + 1 + 7 + `**` = 17 + `**`
- Testing values:
- If `**` = 0 → 17 + 0 = 17 (not divisible by 3)
- If `**` = 1 → 17 + 1 = 18 (divisible by 3)
- **Replace `**` with 1.**

---

**(vi) For `6**1054`:**
- Known digits: 6, 1, 0, 5, 4
- Sum = 6 + 1 + 0 + 5 + 4 + `**` = 16 + `**`
- Testing values:
- If `**` = 0 → 16 + 0 = 16 (not divisible by 3)
- If `**` = 1 → 16 + 1 = 17 (not divisible by 3)
- If `**` = 2 → 16 + 2 = 18 (divisible by 3)
- **Replace `**` with 2.**

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### Final Answers:
1. `27**4` → Replace `**` with 2
2. `53**46` → Replace `**` with 0
3. `8**711` → Replace `**` with 1
4. `62**35` → Replace `**` with 2
5. `234**17` → Replace `**` with 1
6. `6**1054` → Replace `**` with 2