6. Which of the following is not a property of congruence modulo n? A. If a = e mod n and beZ then a + b = c+b mod n. B. If a = c mod n and bEZ then a b =e.b mod n. C. p = p mod n for any prime number p. D. aP-1 = a mod n whenever p is prime.

Q: 3400 m is prime numbers, then compute 27 (mod m) 2. If m is natural numbers, show that a (mod m ) =…

A: Modular arithmetic: a≡b mod c means c divides a-b i.e., c|(a-b). So there exists a constant k such…

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…

A:

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)

A:

Q: If a≡b (mod m) and c ≡d (mod m),show that (b) ac≡bd (mod m).

A: Given : a≡b (mod m) c ≡d (mod m), We have to show  ac≡bd (mod m)

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: 5- If a = b mod(n), then ac = bc mod(n). 6- If a b mod (n), then ak = b*mod(n) for every positive…

A:

Q: Let n be a fixed positive integer greater than 1. If a mod n = a' andb mod n = b', prove that (a +…

A:

Q: Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2q + 1 is also prime. Determine if 2 is a…

A: let p be a prime with p≡3mod 4 and suppose that q=2p+1 is also prime determine if 2 is a square mod…

Q: 13. (a) Prove that for each integer a, if a 0 (mod 7), then a #0 (mod 7).

A:

Q: If a = b (mod m) and a = b (mod n), and gcd (m,n) = 1, then a = b (mod mn) Select one: O True O…

A:

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 r is a primitive root mod a prime p such that p = 3 mod 4. Prove ord,(-r) =

A: Given: p≡3mod 4 To find: ordpp-g=p-12

Q: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

A:

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: For which positive values of k is nk = n (mod 7) for all integers n?

A:

Q: 1. Prove each of the following assertions: (a) If a = b (mod n) and m | n, then a = b (mod m). (b)…

A: Given below the detailed solution

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

A:

Q: Let C={x €Z: x =7 mod 9} and let D={x € Z,x = 1 mod 3). a) List five elements of C and five elements…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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

A:

Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

A:

Q: Give an example to show that a? = b? (mod n) need not imply that a = b mod n). fa=b(mod n) prove…

A:

Q: Prove the following statement or disprove it by giving a counterexample: "For all positive integers…

A: The given statement, "For all positive integer n and for all integers a and b, if a2≡b2modn then…

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…

A:

Q: mod n, then x ≡ 0 mod n or

A:

Q: Prove directly from the definition of congruence modulo n that if a,c, and n are integers,n >1, and…

A: Let a, c and n be integers such that n>1. We have to prove that if a≡cmod n, then a3≡c3mod n…

Q: If a = b (mod n) and b = c (mod n) then which of the following is false. a =c (mod n) a = c (mod n)…

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Q: 30. Prove that any positive integer is congruent to its units digit modulo 10. 31. If a = b (mod n),…

A:

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: 23. Let a,be Z and n eN. If a =b (mod n), then a = ab (mod n). (24, If a b (mod n) and c=d (mod n),…

A:

Q: 4.14. Let a, b, n E Z, where n > 2. Prove that if a = b (mod n), then a2 = b (mod n). 4.15. Let a,…

A: The detailed solution is as follows below:

Q: Prove directly from the definition of congruence modulo n that if a, c, and n are integers, n > 1,…

A:

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: Prove that if n is an odd integer, then n^2 is congruent to 1 modulo 8.

A:

Q: Use the definition of congruence to prove that if j, k, 1, m, nE Z are such that k =1 (mod n) and…

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Q: which one of the following is a congruence ? O23 = 4 mod 5 O 39 = 5 mod 5 O 39 = 10 mod 5 O 29 = 5…

A:

Q: (f) If a = b (mod n), then a* bk (mod n) for any positive integer k.

A:

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: 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…

A:

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…

A:

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: If a and b are integers and n is a positive integer, prove that a mod n =b mod n if and only if n…

A: If a and b are integers and n is a positive integer,    Formula used: a≡b mod n if and only if  a-b…

Q: 12.ifa b(mod n) and b a c(mod n) then ab m be(mod n)

A: We can proove in the following way:-

Q: (a) Consider the following statement: For all n € N, x(x − 1) = 0 mod n Which of "if", "only if",…

A:

Q: Prove the above property. That is, prove that if a mod n = k and b mod n = j, then (a + b) mod n =…

A: If a mod n =k and b mod n=j show that a+b mod n =j+k mod n. Definition: If x mod y=z, then y | x-z.…

Q: 4.18. Let m, ń EN such that m|n. Prove that if a and b are integers such that a = b (mod n), then a…

A: The detailed solution is as follows below:

Q: For any integers m and n, if m mod 11 is 6 and n mod 11 is 7, what is m+n mod 11? 13 9 42

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.