Recall the Modular Equivalence Theorem from Chapter 8: a = b (mod d) meansthat a and b have the same remainder when divided by d. Furthermore, note that if an integer nhas 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 formn = 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 values0, 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 orthe textbook.b) Use part a to prove that a positive integer is divisible by 9 if and only if the sum of its digitsdy + ...+ d, is divisible by 9.

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Description: Recall the Modular Equivalence Theorem from Chapter 8: a = b (mod d) meansthat a and b have the same remainder when divided by d. Furthermore, note that if an integer nhas a remainder of zero when divided by d, then d divides n.Let n be a positive i

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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.

Advanced Engineering Mathematics

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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.

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