Math Problem Statement

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?

Solution

To solve the problem, let's analyze the statements about the smallest number NN 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 NN leaves a remainder of 5 when divided by 6, 8, 12, and 23. This can be mathematically represented as: N5(mod6),N5(mod8),N5(mod12),N5(mod23).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 N5N - 5 is a multiple of 6, 8, 12, and 23. Let M=N5M = N - 5. Then: M is a multiple of 6,8,12, and 23.M \text{ is a multiple of } 6, 8, 12, \text{ and } 23.

2. Finding the Least Common Multiple (LCM)

To find the smallest such MM, we need the least common multiple (LCM) of 6, 8, 12, and 23.

  • The prime factorization of each number is:

    • 6=2×36 = 2 \times 3
    • 8=238 = 2^3
    • 12=22×312 = 2^2 \times 3
    • 23=2323 = 23 (since 23 is a prime number)
  • The LCM is the highest power of each prime factor present in these numbers: LCM(6,8,12,23)=23×3×23.\text{LCM}(6, 8, 12, 23) = 2^3 \times 3 \times 23.

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

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

3. Verifying the Statements

  • Statement 1: "N is not an odd number."

    Checking whether 557 is odd: 557 is odd, so this statement is incorrect.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=17.5 + 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 N5N - 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.

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Math Problem Analysis

Mathematical Concepts

Modular Arithmetic
Number Theory

Formulas

Chinese Remainder Theorem
Calculating LCM

Theorems

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Suitable Grade Level

High School