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6. Let r, denote the least nonnegative residue of 10 (mod 7). Compute ri for i= 1,..., 6. Compute the decimal expansion of the fraction 1/7 without using a calculator. Can you find where the numbers r1, ..., 76 appear in the process of dividing 7 into 1?

6. Let r, denote the least nonnegative residue of 10 (mod 7). Compute ri for i= 1,..., 6. Compute the decimal expansion of the fraction 1/7 without using a calculator. Can you find where the numbers r1, ..., 76 appear in the process of dividing 7 into 1?

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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**Mathematics Problem: Exploring Nonnegative Residues and Decimal Expansions** **Problem Statement:** Let \( r_i \) denote the least nonnegative residue of \( 10^i \) (mod 7). Compute \( r_i \) for \( i = 1, \ldots, 6 \). Compute the decimal expansion of the fraction \( 1/7 \) without using a calculator. Can you find where the numbers \( r_1, \ldots, r_6 \) appear in the process of dividing 7 into 1? **Instructions:** 1. Calculate \( r_1, r_2, \ldots, r_6 \) using the formula \( r_i = 10^i \mod 7 \). 2. Determine the decimal expansion of \( 1/7 \). 3. Analyze how the values \( r_1, \ldots, r_6 \) are involved in the division process. **Concepts Covered:** - Modular arithmetic - Least nonnegative residue - Division and decimal expansion without a calculator This exercise combines concepts of modular arithmetic with practical computation of decimal expansions, enhancing your understanding of both theoretical and applied mathematics.
Transcribed Image Text:**Mathematics Problem: Exploring Nonnegative Residues and Decimal Expansions** **Problem Statement:** Let \( r_i \) denote the least nonnegative residue of \( 10^i \) (mod 7). Compute \( r_i \) for \( i = 1, \ldots, 6 \). Compute the decimal expansion of the fraction \( 1/7 \) without using a calculator. Can you find where the numbers \( r_1, \ldots, r_6 \) appear in the process of dividing 7 into 1? **Instructions:** 1. Calculate \( r_1, r_2, \ldots, r_6 \) using the formula \( r_i = 10^i \mod 7 \). 2. Determine the decimal expansion of \( 1/7 \). 3. Analyze how the values \( r_1, \ldots, r_6 \) are involved in the division process. **Concepts Covered:** - Modular arithmetic - Least nonnegative residue - Division and decimal expansion without a calculator This exercise combines concepts of modular arithmetic with practical computation of decimal expansions, enhancing your understanding of both theoretical and applied mathematics.
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