Every digit of a natural number is either 3 or 4. The number is divisible by both 3 and 4. What is the smallest such number?

Options

A. 333
B. 444
C. 44
D. 4444

Answer

To find the smallest natural number composed only of the digits 3 and 4 that is divisible by both 3 and 4, we can follow these steps:

### Step 1: Understand the divisibility rules
- A number is divisible by **3** if the sum of its digits is divisible by 3.
- A number is divisible by **4** if the number formed by its last two digits is divisible by 4.

### Step 2: List possible combinations of digits
Since every digit of the number can only be 3 or 4, we will start with the smallest combinations:
- 3
- 4
- 33
- 34
- 43
- 44
- 333
- 334
- 343
- 344
- 433
- 434
- 444

### Step 3: Check each combination for divisibility by 3 and 4

1. **Single digits:**
- **3**: Sum = 3 (divisible by 3), last two digits = 3 (not divisible by 4) → Not valid
- **4**: Sum = 4 (not divisible by 3), last two digits = 4 (divisible by 4) → Not valid

2. **Two digits:**
- **33**: Sum = 6 (divisible by 3), last two digits = 33 (not divisible by 4) → Not valid
- **34**: Sum = 7 (not divisible by 3), last two digits = 34 (divisible by 4) → Not valid
- **43**: Sum = 7 (not divisible by 3), last two digits = 43 (not divisible by 4) → Not valid
- **44**: Sum = 8 (not divisible by 3), last two digits = 44 (divisible by 4) → Not valid

3. **Three digits:**
- **333**: Sum = 9 (divisible by 3), last two digits = 33 (not divisible by 4) → Not valid
- **334**: Sum = 10 (not divisible by 3), last two digits = 34 (divisible by 4) → Not valid
- **343**: Sum = 10 (not divisible by 3), last two digits = 43 (not divisible by 4) → Not valid
- **344**: Sum = 11 (not divisible by 3), last two digits = 44 (divisible by 4) → Not valid
- **433**: Sum = 10 (not divisible by 3), last two digits = 33 (not divisible by 4) → Not valid
- **434**: Sum = 11 (not divisible by 3), last two digits = 34 (divisible by 4) → Not valid
- **444**: Sum = 12 (divisible by 3), last two digits = 44 (divisible by 4) → Valid

### Step 4: Conclusion
The smallest natural number composed only of the digits 3 and 4 that is divisible by both 3 and 4 is **444**.