Let m be a positive integer. Show that a mod m = b mod m if a ≡ b (mod m)
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Let m be a positive integer. Show that a mod m = b mod m if a ≡ b (mod m)
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- 32. Prove that if is an integer, then either or . (Hint: Consider the cases where is even and where is odd.)Label each of the following statements as either true or false. a is congruent to b modulo n if and only if a and b yield the same remainder when each is divided by n.a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)