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4. If I have a group of x soldiers, and I divide them into rows to learn the remainder of x when divided by 8, 9, and 10, I cannot necessarily determine the remainder when x is divided by 720.

4. If I have a group of x soldiers, and I divide them into rows to learn the remainder of x when divided by 8, 9, and 10, I cannot necessarily determine the remainder when x is divided by 720.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Statement:** If I have a group of \( x \) soldiers, and I divide them into rows to learn the remainder of \( x \) when divided by 8, 9, and 10, I *cannot* necessarily determine the remainder when \( x \) is divided by 720. **Explanation:** This statement explores number theory, specifically the Chinese Remainder Theorem. Understanding this theorem requires knowledge of remainders and how they behave under various conditions. **Concepts to Consider:** 1. **Remainder and Modulo Operation:** - The remainder is what's left after division. - For example, dividing \( x \) by 8 gives a remainder represented as \( x \mod 8 \). 2. **Chinese Remainder Theorem (CRT):** - CRT is useful for simultaneous congruences, offering solutions if divisors are pairwise coprime. - In this example, divisors (8, 9, and 10) are used to find \( x \) mod 720. 3. **Coprime Numbers:** - Numbers like 8, 9, and 10 are not pairwise coprime because 8 shares a factor with 10 (both divisible by 2). By dividing \( x \) by 8, 9, and 10, and knowing their remainders, one might assume the possibility of finding a unique remainder for 720 using CRT. However, due to the lack of pairwise coprimeness, this isn't assured, making the determination of \( x \) mod 720 uncertain.
Transcribed Image Text:**Statement:** If I have a group of \( x \) soldiers, and I divide them into rows to learn the remainder of \( x \) when divided by 8, 9, and 10, I *cannot* necessarily determine the remainder when \( x \) is divided by 720. **Explanation:** This statement explores number theory, specifically the Chinese Remainder Theorem. Understanding this theorem requires knowledge of remainders and how they behave under various conditions. **Concepts to Consider:** 1. **Remainder and Modulo Operation:** - The remainder is what's left after division. - For example, dividing \( x \) by 8 gives a remainder represented as \( x \mod 8 \). 2. **Chinese Remainder Theorem (CRT):** - CRT is useful for simultaneous congruences, offering solutions if divisors are pairwise coprime. - In this example, divisors (8, 9, and 10) are used to find \( x \) mod 720. 3. **Coprime Numbers:** - Numbers like 8, 9, and 10 are not pairwise coprime because 8 shares a factor with 10 (both divisible by 2). By dividing \( x \) by 8, 9, and 10, and knowing their remainders, one might assume the possibility of finding a unique remainder for 720 using CRT. However, due to the lack of pairwise coprimeness, this isn't assured, making the determination of \( x \) mod 720 uncertain.
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