m and n be positive integers so that m n. Show for any integer a that the congruence

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Q: Suppose that k and n > k are fixed positive integer. Justify the identity n n + 1 %3D k +1, j=k

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A: Detailed proof is provided in the next step.

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A: Given that for positive integers a,b,c,d,e,f , d=gcda,b,ce=gcda,bf=gcde,c

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Q: 3. Find integers s and t so that 1 7s+ 11t. Show that they are not unique. %3D

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Q: Prove that if a > 0, k > 1, and ak -1 is prime, then a = 2 and k is prime. %3D

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Q: Let m >1 be an integer, and let R = Z/mZ. Prove that R is nice if and only m is prime. Proof. ...

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A: If m an n are both even  let m= 2a and n= 2b   then  5m+7n=5(2a)+7(2b)=10a+14b=25a+7b therefore,…

Q: 2. Show that if a and b are positive integers and alb, then a < b. 3. Show that if a and b are…

A: Basic facts about division will be used in solving this problem.

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Q: Let m, n be positive integers. If m n, prove that 5m - 1 5" - 1

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