Show that there are no integers a, b such that a² + 2ab+26² = 100000000003. Hint. Use mod 4 congruences.
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- 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.)In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and to determine whether there are solutions. If there are, find incongruent solutions modulo. 50.