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3. Use your result above to show that the number 1,111,111,113 is divisible by 3 while the number 111,111,113 is not.

3. Use your result above to show that the number 1,111,111,113 is divisible by 3 while the number 111,111,113 is not.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
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In base 10 representation, every integer n is the sum of some multiples of the powers of 10. For example, if n = 12345, then n = 1· 104 +2 · 10³ + 3 · 10² + 4 · 10' +5· 10°. In general, if the digits of the base-10 representation of n form the string apar-1 ·.. ao, then >a; · 10'. (1) n = i=0 1. Show that an integer n is divisible by 2 if and only if its last digit is even. (Hint: For each i > 1 in Eq. (1), 10² = 0 (mod 2). When ao is even, what is it congruent to modulo 2?) 2. An integer n is divisible by 3 if and only if the sum of its digits is divisible by 3. (Hint: Use modular arithmetic. What is 10 congruent to modulo 3?) 3. Use your result above to show that the number 1,111,111,113 is divisible by 3 while the number 111,111,113 is not. 4. Solve the congruence 34x = 77 (mod 89). 5. What is the original message encrypted using the RSA system with n = 5 · 7 and e = 5 if the encrypted message is 33 14 21 08 09 18?
Transcribed Image Text:In base 10 representation, every integer n is the sum of some multiples of the powers of 10. For example, if n = 12345, then n = 1· 104 +2 · 10³ + 3 · 10² + 4 · 10' +5· 10°. In general, if the digits of the base-10 representation of n form the string apar-1 ·.. ao, then >a; · 10'. (1) n = i=0 1. Show that an integer n is divisible by 2 if and only if its last digit is even. (Hint: For each i > 1 in Eq. (1), 10² = 0 (mod 2). When ao is even, what is it congruent to modulo 2?) 2. An integer n is divisible by 3 if and only if the sum of its digits is divisible by 3. (Hint: Use modular arithmetic. What is 10 congruent to modulo 3?) 3. Use your result above to show that the number 1,111,111,113 is divisible by 3 while the number 111,111,113 is not. 4. Solve the congruence 34x = 77 (mod 89). 5. What is the original message encrypted using the RSA system with n = 5 · 7 and e = 5 if the encrypted message is 33 14 21 08 09 18?
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