Prove the following:   a. If n is an integer and n + 3 is even, then n is odd by using a proof by contraposition. b.  If a ≡ b (mod m) and b ≡ c (mod m) where a and b are non-zero integers and m is a positive integer, then a ≡ c (mod m).

Q: Let n be a positive integer. Prove 41+3n mod 9.

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Q: Let n E Z+, n > 2. Show that n is prime if and only if (n – 2)! = 1 (mod n).

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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: 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: Let x be an integer. Prove that, if x mod 9 = 5, then 9|(x^2-2x+3)

A: Given: x mod 9=5 then, 9|x2-2x+3|

Q: 1. Use Euler's theorem to establish the following: (a) For any integer a, a [Hint: 1729 = 7. 13 ·…

A: As per the question we will use the following Euler's theorem : -  aφ(n) ≡  1 (mod n) Where φ(n) is…

Q: For any integer a, prove that a = 0 or 1 (mod 3).

A: Every number is either of the forms 3k, 3k−1, 3k+1 where k is an integer

Q: Suppose that a and b are integers such that a = 45 (mod 71) and b = 53 (mod 71). Find an integer c…

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Q: Let x be an integer. Prove that, if mod3=2, then 3|(x2-1).

A: I hove used the theorem on congruent

Q: Prove that for any integer k, at least one of k, k + 2, or k + 4 is divisible by 3. (Hint: consider…

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Q: For any integer a, prove that a=0 or 1 (mod 3).

A: The solution is given as follows

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

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Q: prove, for all integers x, if x mod 3 does not equal 0, then 3 | (x2 +2)

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

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Q: Establish that if a is an odd integer, then 2" aʻ = 1 (mod 2" *2) for any n 2 1.

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Q: Prove a2 = c² (mod n), assuming that a, c and n are integers with n > 1 and that a = c (mod n)

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

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Q: Let a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n).

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Q: Establish that if a is an odd integer, then 2" а = 1 (mod 2" +2) for any n 2 1.

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Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

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Q: Prove that for every integer n, n3 mod 6 = n mod 6.

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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: If m and n are relatively prime integers, then: a) mº(n) + nº(m) = 1 mod(mn) b) т+nmod (mn) c) -1…

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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: For any integer a, a37 = ? mod 1729

A: Option b is correct answer

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: 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: 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: 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: Establish that if a is an odd integer, then 2" a“ = 1 (mod 2" *2) for any n 2 1.

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Q: Prove that if n is an odd positive integer, then n² = 1 (mod 8).

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

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Q: Let n E Z. Prove that 3n # 1 (mod 9).

A: Proof by contradiction: Direct approaches can be difficult (or impossible) to use to prove a…

Q: 2222 (mod 13) is equal to O 1 (mod 13) O 12 (mod 13) O 2 (mod 13) None of the mentioned O 6 (mod 13)

A: Properties of modulo function  a≡b(mod n ) then an≡ bn(mod n) a≡b(mod n ) then ac≡ bc(mod n)…

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

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Q: Let n E Z. Prove that if 5n2 +3 =0 (mod 8), then 4 n.

A: In this question  we will be using contradiction concept so assume it is true statement then we…

Q: For any integer n, either n2 mod 4 = 0, or n² mod 4 = 1.

A: For any integer n, we prove that either n2 mod4=0,  or n2 mod4=1

Q: Establish that if a is an odd integer, then 2" = 1 (mod 2" +2) for any n 2 1.

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Q: (a) Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1.…

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Q: Given that a, b, c,m are integers with m > 0. If a = b (mod m) then prove that ас3 bе (mod m)

A: we have to use the only definition of congruence and the definition of divisibility

Q: Prove: for any integer x, if x mod 3 is not 0, then (x2−3x+ 1) mod 3 = 2.

A: To prove :     for any integer x, if xmod 3≢0, then x2-3x+1mod 3=2   Pre-requisite : 1. a≡bmod n…

Q: 1. Prove each of the following assertions: (c) If a = b (mod n) and the integers a, b, n are all…

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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: Prove, for all integers x, if x mod 3 is not 0, then 3|(x2+2).

A: Congruence Modulo Relation on Integers: Let a, b∈ℤ and let n be any fixed positive integer. Then…

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

A: Looking at the definition of congurence we can easily do these two problems and we need some algebra…