Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n) and c = d( mod n) then a – c = b - d ( mod n).

Q: Prove if a=b mod m then a^n=b^n mod m use Math induction

A: We have to prove that if a=b mod m then a^n=b^n mod m

Q: 6. Let m, n EN with m, n > 2, and let a, b E Z. Prove that a = b (mod mn) implies a = b (mod m) and…

A: (a) a = b (mod mn)       => there exists some integer k such that                    a = b +…

Q: Show that if a and b are integers and m and n are positive integers such that n|m, then If a = b(mod…

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Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

Q: Prove that if a = b (modn) and m/n, then a = b (modm).

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Q: Prove the following statement or disprove by giving a counterexample "for all positive intergers n…

A: We have to prove or disprove the following statement For all positive integer n and for all integer…

Q: For any a, b∈Z and positive integer n, we say that a is congruent to b module N, written a≡b (mod…

A: Just use the mentioned definition of congruence modulo.

Q: If a = b (mod m) and a = b (mod n) then a = b (mod mn)

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Q: Use induction to prove that for every n E N, we have 5" = 1+ 4n (mod 8)

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Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

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Q: consider the statement: for all n∈N , x∈Z , x^2 ≡ 1 mod n (        ) x ≡ 1 mod n or x≡ -1 mod n a)…

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Q: Prove the statement by contradiction.

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Q: Suppose m, n e Z† are relatively prime. Prove that for all a, b e Z, a = b (mod mn) iff a = b (mod…

A: The congruence relation a≡b mod n implies that a-b is a multiple of n. That is n| a-b or a=kn+b If…

Q: Let a, b E Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

A: The given problem is to prove the following identities, we have to use basic identities of modular…

Q: 24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

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Q: Prove or disprove If ac = bc(mod m), then a = b(mod m). If n| m and a = b(mod m), then a = b(mod n).…

A: Hello. Since you have posted multiple questions and not specified which question needs to be solved,…

Q: Prove or disprove If ac = bc(mod m), then a = b(mod m). If n | m and a = b(mod m), then a = b(mod…

A: According to the given information, it is required to prove or disprove the given statement:

Q: Suppose m,n e Z+ are relatively prime. Prove that for all a,b e Z, a = b (mod mn) iff a = b (mod m)…

A: Given: a,b∈Z and  m,n∈Z+ are relatively prime .

Q: a. Let m be an integer such that m mod 5 = 2, what is 8m mod 5? b. Let n be an integer such that n…

A: Consider the given information: Let m be an integer such that m mod5 = 2. Then m=5k+2   where…

Q: Let n ∈ Z. Prove that if n^2 + 2n −4 doesnt equal 1 (mod 5), then 5 doesnt divide n.

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Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

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Q: If n| m and a = b(mod m), then a = b(mod n).

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Q: Q1 Prove or disprove i. If ac = bc(mod m), then a = b(mod m).

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Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

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Q: Prove that for any prime, p = 1 (mod 4) there are exactly (p-1)/4 incongruent values of a 0 (mod p)…

A: By theorem (a,p)=1 then xn≡a modp has g.c.d(n,p-1) solution or no solution according as…

Q: 20. Are the following statements true or false? Either prove the statement is true or provide a…

A: PART A For the integers a and b, a.b=0(mod 6) This implies that either a=0(mod6) or b=0(mod6). So,…

Q: (c) For all integers a and b, if a · b = 1 (mod 6), then a =1 (mod 6) or b = 1 (mod 6). (d) For all…

A: (c) The statement is false. For example take a=5, b=5 ab=2525≡1 mod(6) Hence ab≡1 mod(6) But a≡5…

Q: Prove that min (a) If a = b (mod n) and mln ,then a = b (mod m) (b) If a = b (mod n) and c =d (mod…

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Q: Suppose that each of a, b, and c is an integer and m is a positive integer. Show that if a = b(mod…

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Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

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Q: Let a be an integer and let n e N. (a) Prove that if a = 0 (mod n), then n | a.

A: By the definition if congruence let a, b and n are integer, with n >0, then a is congruent to b…

Q: Show that: if a = b(mod n) and C = d\modn), then a +c =b+d\mod n)!

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Q: Show that if a is an even integer then a² = 0 (mod 4) and if a is odd integer then a² = 1 (mod 4)

A: We know that an even number is a multiple of 2 and we write an even number k as k=2m for some…

Q: Suppose that a and b are integers, a = 4 (mod 13), and b = 9 (mod 13). Find the integer c with 0 < c…

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Q: Let n be an integer. Prove that, if n mod6 = 2, then n2mod12=4

A: We apply concept of modulus here to solve this.

Q: show that If n is a positive integer and a is any integer prime to n, then a ln) = 1(mod n), where…

A: It's called Euler's generalization to Fermats theorem. I have used integer m instead of n.

Q: Give an example to show that a ^2 ≡ b^2 (mod m) need not imply that a≡ b (mod m)

A: Here I listed given example of above statement.

Q: Prove that if a = b mod n and c= d mod n, then ac bd mod n.

A: Given if a≡bmodn and c≡dmodn We have to show that ac≡bdmodn. Now, a≡bmodn…

Q: Prove that for each integer a, if a not equal to 0 (mod 7), then a2 not equal to 0 (mod 7)

A: Consider the provided question,

Q: Prove the following:   a. If n is an integer and n + 3 is even, then n is odd by using a proof by…

A: Remark: Proof by contrapositive is based on the fact that an implication is equivalent to its…

Q: Prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)

A: We have to prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)

Q: Let a, b, and c be integers and n a natural number. If a ≡ b mod n then a+c ≡ b+c mod n. How do I…

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Q: Show that if a = b mod m and b = c mod m, then we have a = c mod m. Use the definition of congruence…

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Q: Prove or disprove: for any integers a, b, c, d, if a= c (mod n) and b= d (mod n), then ab = cd (mod…

A: Given: For any integers a, b, c, d if a≡cmod n and b≡dmod n To show: ab≡cd (mod n) To say that a is…

Q: Show that if a is an even integer, then a =0 (mod 4), and if a is an odd integer, then a =1 (mod 4).

A: Suppose a is an even integer Then a=2m for some integer m a2=(2m)2 a2=4m2Clearly 4|4so 4|4m2so…

Q: Prove or disprove: If a = b(mod n²) then a = b(mod n)

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Q: For which primes p > 7 is 7 a square mod p? Your answer should be of the form p = {a1, a2,... , a}…

A: Given: p&gt;7 ,If 7 is a square mod p.

Q: Suppose that Bob has used two distinct primes p and q to create the public modulus n = pq and public…

A: RSA algorithm: 1. Choose two distinct prime numbers p and q. Take n = pq. 2. Compute λ(n), where λ…