Replace * by the smallest digit so that:
2*3614 is divisible by 6

Answer

To solve the problem of replacing * in the number 2*3614 so that it becomes divisible by 6, we need to follow these steps:

### Step 1: Understand the divisibility rule for 6
A number is divisible by 6 if it is divisible by both 2 and 3.

### Step 2: Check divisibility by 2
For a number to be divisible by 2, its last digit must be even. In our case, the last digit of 23614 is 4, which is even. Therefore, any number formed by replacing * will still be divisible by 2.

### Step 3: Check divisibility by 3
For a number to be divisible by 3, the sum of its digits must be divisible by 3. We will calculate the sum of the digits in 2*3614.

Let’s denote the digit we replace * with as x. The digits are 2, x, 3, 6, 1, and 4.

The sum of the digits is:
\[ 2 + x + 3 + 6 + 1 + 4 = 16 + x \]

### Step 4: Find the smallest digit x such that (16 + x) is divisible by 3
We will check each digit from 0 to 9 to find the smallest x.

- If \( x = 0 \): \( 16 + 0 = 16 \) (not divisible by 3)
- If \( x = 1 \): \( 16 + 1 = 17 \) (not divisible by 3)
- If \( x = 2 \): \( 16 + 2 = 18 \) (divisible by 3)
- If \( x = 3 \): \( 16 + 3 = 19 \) (not divisible by 3)
- If \( x = 4 \): \( 16 + 4 = 20 \) (not divisible by 3)
- If \( x = 5 \): \( 16 + 5 = 21 \) (divisible by 3)
- If \( x = 6 \): \( 16 + 6 = 22 \) (not divisible by 3)
- If \( x = 7 \): \( 16 + 7 = 23 \) (not divisible by 3)
- If \( x = 8 \): \( 16 + 8 = 24 \) (divisible by 3)
- If \( x = 9 \): \( 16 + 9 = 25 \) (not divisible by 3)

### Step 5: Identify the smallest digit
From the checks above, the smallest digit x that makes \( 16 + x \) divisible by 3 is 2.

### Conclusion
Thus, the smallest digit to replace * in 2*3614 to make it divisible by 6 is **2**.

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