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 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 leaves a remainder of 5 when divided by 6, 8, 12, and 23. This can be mathematically represented as:
This implies that is a multiple of 6, 8, 12, and 23. Let . Then:
2. Finding the Least Common Multiple (LCM)
To find the smallest such , we need the least common multiple (LCM) of 6, 8, 12, and 23.
-
The prime factorization of each number is:
- (since 23 is a prime number)
-
The LCM is the highest power of each prime factor present in these numbers:
Calculating this gives:
Thus, , so the smallest .
3. Verifying the Statements
-
Statement 1: "N is not an odd number."
Checking whether 557 is odd:
-
Statement 2: "Sum of digits of N is 16."
Calculating the sum of the digits of 557: 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:
- How do we compute the least common multiple (LCM) of several numbers?
- Why does need to be a multiple of the LCM of 6, 8, 12, and 23?
- What methods can be used to verify whether a number is odd or even?
- How can you quickly calculate the sum of digits of a number?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Modular Arithmetic
Number Theory
Formulas
Chinese Remainder Theorem
Calculating LCM
Theorems
-
Suitable Grade Level
High School
Related Recommendation
Find the Least Number with Remainder 2 When Divided by 3, 4, 5, and 6
Finding the Smallest Number Satisfying Multiple Congruences Using the Chinese Remainder Theorem
Find the Least Number with Remainder 5 When Divided by 16, 28, 40, and 77
Find the Smallest Number That Leaves a Remainder of 8 When Divided by 15, 20, or 35
Solve Modular Arithmetic Problem using Chinese Remainder Theorem