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30. Prove that any positive integer is congruent to its units digit modulo 10. 31. If a = b (mod n), prove that a" = bm (mod n) for every positive integer m. 32. Prove that if m is an integer, then either m2 = 0 (mod 4) or m2 = 1 (mod 4) Consider the cases where m is even and where m is odd.)

30. Prove that any positive integer is congruent to its units digit modulo 10. 31. If a = b (mod n), prove that a" = bm (mod n) for every positive integer m. 32. Prove that if m is an integer, then either m2 = 0 (mod 4) or m2 = 1 (mod 4) Consider the cases where m is even and where m is odd.)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.5: Congruence Of Integers
Problem 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is...
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Please help with number 31. 

30. Prove that any positive integer is congruent to its units digit modulo 10. 31. If a = b (mod n), prove that a" = bm (mod n) for every positive integer m. 32. Prove that if m is an integer, then either m2 = 0 (mod 4) or m² = 1 (mod 4) Consider the cases where m is even and where m is odd.)
Transcribed Image Text:30. Prove that any positive integer is congruent to its units digit modulo 10. 31. If a = b (mod n), prove that a" = bm (mod n) for every positive integer m. 32. Prove that if m is an integer, then either m2 = 0 (mod 4) or m² = 1 (mod 4) Consider the cases where m is even and where m is odd.)
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