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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.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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