Consider the following statements about the smallest number N when divided by 6, 8,
12 and 23 leaves reminder of 5?
Statement-1: N is not an odd number.
Statement-2: Sum of digits of N is 16.
Which one of the following is/are correct?

To solve the problem, let's analyze the statements about the smallest number \( N \) that leaves a remainder of 5 when divided by 6, 8, 12, and 23.

### Step-by-Step Solution:

#### 1. Understanding the Problem
The problem states that \( N \) leaves a remainder of 5 when divided by 6, 8, 12, and 23. This can be mathematically represented as:
\[
N \equiv 5 \pmod{6}, \quad N \equiv 5 \pmod{8}, \quad N \equiv 5 \pmod{12}, \quad N \equiv 5 \pmod{23}.
\]

This implies that \( N - 5 \) is a multiple of 6, 8, 12, and 23. Let \( M = N - 5 \). Then:
\[
M \text{ is a multiple of } 6, 8, 12, \text{ and } 23.
\]

#### 2. Finding the Least Common Multiple (LCM)
To find the smallest such \( M \), we need the least common multiple (LCM) of 6, 8, 12, and 23.

- The prime factorization of each number is:
  - \( 6 = 2 \times 3 \)
  - \( 8 = 2^3 \)
  - \( 12 = 2^2 \times 3 \)
  - \( 23 = 23 \) (since 23 is a prime number)

- The LCM is the highest power of each prime factor present in these numbers:
  \[
  \text{LCM}(6, 8, 12, 23) = 2^3 \times 3 \times 23.
  \]

Calculating this gives:
\[
\text{LCM} = 8 \times 3 \times 23 = 24 \times 23 = 552.
\]

Thus, \( M = 552 \), so the smallest \( N = M + 5 = 552 + 5 = 557 \).

#### 3. Verifying the Statements

- **Statement 1: "N is not an odd number."**

  Checking whether 557 is odd:
  \[
  557 \text{ is odd, so this statement is incorrect.}
  \]

- **Statement 2: "Sum of digits of N is 16."**

  Calculating the sum of the digits of 557:
  \[
  5 + 5 + 7 = 17.
  \]
  This statement is also incorrect.

### Conclusion

Both statements are incorrect.

**Answer: None of the statements are correct.**

---

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do we compute the least common multiple (LCM) of several numbers?
2. Why does \(N - 5\) need to be a multiple of the LCM of 6, 8, 12, and 23?
3. What methods can be used to verify whether a number is odd or even?
4. How can you quickly calculate the sum of digits of a number?
5. What other problems can be solved using the concept of LCM and modular arithmetic?

**Tip:** Understanding modular arithmetic and LCM is crucial for solving problems related to divisibility and remainders.

Learn how to solve a modular arithmetic problem to find the smallest number N that leaves a remainder of 5 when divided by 6, 8, 12, and 23. Includes step-by-step solution using modular arithmetic and LCM calculation. Suitable for high school students studying number theory.

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