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Let a = 25, b = 19, and n = 3. a. Verify that 3|(25 – 19). b. Explain why 25 = 19 (mod 3). c. What value of k has the property that %3D 25 = 19+3k? d. What is the (nonnegative) remainder obtained when 25 is divided by 3? When 19 is divided by 3? e. Explain why 25 mod 3 = 19 mod 3.

Let a = 25, b = 19, and n = 3. a. Verify that 3|(25 – 19). b. Explain why 25 = 19 (mod 3). c. What value of k has the property that %3D 25 = 19+3k? d. What is the (nonnegative) remainder obtained when 25 is divided by 3? When 19 is divided by 3? e. Explain why 25 mod 3 = 19 mod 3.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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**Understanding Modular Arithmetic and Divisibility** Let \( a = 25 \), \( b = 19 \), and \( n = 3 \). **a. Verify that \( 3 \mid (25 - 19) \).** To verify this, calculate \( 25 - 19 = 6 \). Since 6 is divisible by 3 (i.e., \( 6 \div 3 = 2 \)), it confirms that \( 3 \mid 6 \). **b. Explain why \( 25 \equiv 19 \pmod{3} \).** In modular arithmetic, two numbers are equivalent when they have the same remainder when divided by the modulus. For \( 25 \equiv 19 \pmod{3} \), we need to check the remainders: - The remainder of \( 25 \div 3 \) is 1. - The remainder of \( 19 \div 3 \) is also 1. Since both remainders are the same, \( 25 \equiv 19 \pmod{3} \). **c. What value of \( k \) has the property that \( 25 = 19 + 3k \)?** To find \( k \), rearrange the equation: \( 25 - 19 = 3k \) \( 6 = 3k \) \( k = 2 \). Thus, \( k = 2 \). **d. What is the (nonnegative) remainder obtained when 25 is divided by 3? When 19 is divided by 3?** - The remainder of \( 25 \div 3 \) is 1. - The remainder of \( 19 \div 3 \) is 1. **e. Explain why \( 25 \mod 3 = 19 \mod 3 \).** Both \( 25 \) and \( 19 \) have the same remainder when divided by 3, which is 1. Therefore, \( 25 \mod 3 = 19 \mod 3 \) because they leave the same remainder. This explains their equivalence in modular arithmetic.
Transcribed Image Text:**Understanding Modular Arithmetic and Divisibility** Let \( a = 25 \), \( b = 19 \), and \( n = 3 \). **a. Verify that \( 3 \mid (25 - 19) \).** To verify this, calculate \( 25 - 19 = 6 \). Since 6 is divisible by 3 (i.e., \( 6 \div 3 = 2 \)), it confirms that \( 3 \mid 6 \). **b. Explain why \( 25 \equiv 19 \pmod{3} \).** In modular arithmetic, two numbers are equivalent when they have the same remainder when divided by the modulus. For \( 25 \equiv 19 \pmod{3} \), we need to check the remainders: - The remainder of \( 25 \div 3 \) is 1. - The remainder of \( 19 \div 3 \) is also 1. Since both remainders are the same, \( 25 \equiv 19 \pmod{3} \). **c. What value of \( k \) has the property that \( 25 = 19 + 3k \)?** To find \( k \), rearrange the equation: \( 25 - 19 = 3k \) \( 6 = 3k \) \( k = 2 \). Thus, \( k = 2 \). **d. What is the (nonnegative) remainder obtained when 25 is divided by 3? When 19 is divided by 3?** - The remainder of \( 25 \div 3 \) is 1. - The remainder of \( 19 \div 3 \) is 1. **e. Explain why \( 25 \mod 3 = 19 \mod 3 \).** Both \( 25 \) and \( 19 \) have the same remainder when divided by 3, which is 1. Therefore, \( 25 \mod 3 = 19 \mod 3 \) because they leave the same remainder. This explains their equivalence in modular arithmetic.
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