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a) Prove that for all integers n > 0, 10" = 1(mod 9). You may refer to theorems from class or the textbook.

a) Prove that for all integers n > 0, 10" = 1(mod 9). You may refer to theorems from class or the textbook.

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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Recall the Modular Equivalence Theorem from Chapter 8: a = b (mod d) means that a and b have the same remainder when divided by d. Furthermore, note that if an integer n has a remainder of zero when divided by d, then d divides n. Let n be a positive integer. The decimal representation of n is an expression of the form n = d,10k + dx-110"-1 + ... + d, 10' + d,10° Where the coefficients dr, ., do are called the digits of n. The digits take on the values 0, 1,2, .,9 and the first digit d, is nonzero. .... a) Prove that for all integers n > 0, 10" = 1(mod 9). You may refer to theorems from class or the textbook. b) Use part a to prove that a positive integer is divisible by 9 if and only if the sum of its digits dy + ...+ d, is divisible by 9.
Transcribed Image Text:Recall the Modular Equivalence Theorem from Chapter 8: a = b (mod d) means that a and b have the same remainder when divided by d. Furthermore, note that if an integer n has a remainder of zero when divided by d, then d divides n. Let n be a positive integer. The decimal representation of n is an expression of the form n = d,10k + dx-110"-1 + ... + d, 10' + d,10° Where the coefficients dr, ., do are called the digits of n. The digits take on the values 0, 1,2, .,9 and the first digit d, is nonzero. .... a) Prove that for all integers n > 0, 10" = 1(mod 9). You may refer to theorems from class or the textbook. b) Use part a to prove that a positive integer is divisible by 9 if and only if the sum of its digits dy + ...+ d, is divisible by 9.
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