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Determine the possibilities for the final digit of a sixth power of an integer.

Determine the possibilities for the final digit of a sixth power of an integer.

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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1. Only one prime of the form 1+4n exists. Determine this prime number and prove it's the only one of this particular form. Hint: Research Sophie Germain's Identity to factor 1+4n4. ad bowollal ng (0 be) + 2. Let p be a prime number such that p> 5. Prove that p2 -1 = 0 (mod 24). odo eit l 3. Let q be a prime and n EN such that 1<n<q. Prove that q cheofla edt ml o bo 4. Use the theory of congruences to verify that 25| (27+4 +33n+2 – 53n+6) for all n eN 5. Using congruence theory (not brute force), find all solutions to the following linear congruence: 8x + 9y = 10 (mod 11) at uq n odT teeW tuo msd hio g ao clett a bodder eio I Sol ar nd blon sond U sto bas , at bx b ed . blog sr etth bing ls d a adpah o i obloeb 6. Determine the possibilities for the final digit of a sixth power of an integer. d to pahvol bio a (od) eg ) S y be h 7. Prove that if n is an odd positive integer or divisible by 4, then 13 + 23 + 33 + .. + (n – 1)3 = 0 (mod n) Is the statement true if n is even but not divisible by 4?
Transcribed Image Text:1. Only one prime of the form 1+4n exists. Determine this prime number and prove it's the only one of this particular form. Hint: Research Sophie Germain's Identity to factor 1+4n4. ad bowollal ng (0 be) + 2. Let p be a prime number such that p> 5. Prove that p2 -1 = 0 (mod 24). odo eit l 3. Let q be a prime and n EN such that 1<n<q. Prove that q cheofla edt ml o bo 4. Use the theory of congruences to verify that 25| (27+4 +33n+2 – 53n+6) for all n eN 5. Using congruence theory (not brute force), find all solutions to the following linear congruence: 8x + 9y = 10 (mod 11) at uq n odT teeW tuo msd hio g ao clett a bodder eio I Sol ar nd blon sond U sto bas , at bx b ed . blog sr etth bing ls d a adpah o i obloeb 6. Determine the possibilities for the final digit of a sixth power of an integer. d to pahvol bio a (od) eg ) S y be h 7. Prove that if n is an odd positive integer or divisible by 4, then 13 + 23 + 33 + .. + (n – 1)3 = 0 (mod n) Is the statement true if n is even but not divisible by 4?
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