16. Prove that an integer (a,-1an-2 · . . ao)10 is divisible by 11 if and only if ao + a2 + a4 + . . . = aj + a3 + a5 + · ·. (mod 11). [Hint: 10 = -1 (mod 11).] %3D
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- 30. Prove that any positive integer is congruent to its units digit modulo .25. Complete the proof of Theorem : If and is any integer, then .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.)