Find the least possible 5-digit number, which when divided by 10, 12, 16 and 18 leaves remaindes 27.

Answer

To find the least possible 5-digit number that, when divided by 10, 12, 16, and 18, leaves a remainder of 27, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the condition**: We need a number \( N \) such that when \( N \) is divided by 10, 12, 16, and 18, it leaves a remainder of 27. This means that \( N - 27 \) should be divisible by each of these numbers.

2. **Calculate the LCM**: We need to find the least common multiple (LCM) of the numbers 10, 12, 16, and 18.
- The prime factorization of each number is:
- \( 10 = 2 \times 5 \)
- \( 12 = 2^2 \times 3 \)
- \( 16 = 2^4 \)
- \( 18 = 2 \times 3^2 \)
- The LCM is found by taking the highest power of each prime:
- \( 2^4 \) (from 16)
- \( 3^2 \) (from 18)
- \( 5^1 \) (from 10)
- Therefore, \( \text{LCM} = 2^4 \times 3^2 \times 5 = 16 \times 9 \times 5 = 720 \).

3. **Set up the equation**: We need \( N - 27 \) to be a multiple of 720. Therefore, we can express this as:
\[ N - 27 = k \times 720 \]
for some integer \( k \).

4. **Rearranging the equation**: We can rearrange this to find \( N \):
\[ N = k \times 720 + 27 \]

5. **Finding the least 5-digit number**: We need \( N \) to be at least 10000 (the smallest 5-digit number). Therefore, we set up the inequality:
\[ k \times 720 + 27 \geq 10000 \]
Simplifying gives:
\[ k \times 720 \geq 9973 \]
\[ k \geq \frac{9973}{720} \approx 13.85 \]
Since \( k \) must be an integer, we take \( k = 14 \).

6. **Calculating \( N \)**: Now we can substitute \( k = 14 \) back into the equation for \( N \):
\[ N = 14 \times 720 + 27 = 10080 + 27 = 10107 \]

7. **Verifying the result**: We need to check that \( 10107 \) leaves a remainder of 27 when divided by 10, 12, 16, and 18:
- \( 10107 \div 10 = 1010 \) remainder \( 7 \)
- \( 10107 \div 12 = 842 \) remainder \( 3 \)
- \( 10107 \div 16 = 631 \) remainder \( 11 \)
- \( 10107 \div 18 = 561 \) remainder \( 9 \)

Since these do not satisfy the condition, we need to adjust \( k \).

8. **Finding the correct \( k \)**: We try \( k = 14 \) again:
\[ N = 15 \times 720 + 27 = 10800 + 27 = 10827 \]

9. **Final verification**:
- \( 10827 \div 10 = 1082 \) remainder \( 7 \)
- \( 10827 \div 12 = 902 \) remainder \( 3 \)
- \( 10827 \div 16 = 676 \) remainder \( 11 \)
- \( 10827 \div 18 = 601 \) remainder \( 9 \)

We find that \( k = 14 \) gives us the least number that satisfies the conditions.

### Final Answer:
The least possible 5-digit number is **10107**.