Show that if a, b, and m are integers such that m > 2 and a = b (mod m), then ged(a, m) = gcd(b, m).

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: Let x E Z and y E Z. Prove the following statement: If y = 2 mod 4, then xy 1 mod 4.

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Q: Let a, b, n be positive integers. Prove that if a = b (mod n), then GCD(a, n) = GCD(b,n).

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Q: Let m be a positive integer. Show that a mod m = b mod m if a ≡ b (mod m)

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Q: Prove for any integers a and b, if (a3+b3) mod 3 is not 0, then (a+b) mod 3 is not 0. (Hint: recall…

A: Congruence Modulo Relation on Integers: Let a, b∈ℤ and let m be a fixed positive integer.  Then 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: 12. Prove that if gcd(a, n) { b, then ax = b (mod n) has no solutions.

A: To prove the given statement, it is equivalent to prove its contrapositive statement which is "if…

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: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

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Q: a) Find integers a and b such that a + b =a-b (mod 5).

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Q: 1. Let a, b e Z. Prove that if a = 0 (mod 5) and b = 2 (mod 5), then a² + b² = 4 (mod 5).

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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: Let n and a be positive integers and let d = gcd(a, n). Show that the equation ax mod n = 1 has a…

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Q: Prove that if a and b are integers, then (a^5 + b^5) is congruent to a^5 + b^5 modulo 5.

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Q: Prove that ifn > 1 and a > 0 are integers, and d = GCD(a,n), then the additive order of a modulo m,…

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Q: Assuming Wilson's Theorem, prove that if p prime and (a, p) = 1, then aP-1 = 1 mod p.

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Q: Prove that if p is a prime and a is any integer so that p does not divide a, then the additive order…

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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 a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n). (Hint: use the…

A: The greatest common divisor (gcd) of two or more numbers is the greatest common factor number that…

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: Let a, b, c, m be integers with m > 2. Prove that if ac ≡ bc mod m and d = gcd(m, c) , then a ≡ b…

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Q: 4. Exercise $2.5 #58a. Let a and b be integers not divisible by the prime mumber p. (a) If aP = mod…

A: Given the data

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: Show that for any positive integer x, at least one of x, x+ 2, or x+ 4 is divisible by 3

A: The objective is to show that for any positive integer x at one of x, x+ 2, or x+ 4 is divisible by…

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: Find a number a such that 70a = 1 mod 17.

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Q: Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).

A: we will prove that gcd (a, n) = 1, which  means that a and n are  prime. Let m= gcd (a, n) = gcd (a…

Q: If gcd(a, c) = 1 and gcd(b, c) = 1, prove that gcd(ab, c) = 1.

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Q: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡−1 mod 7

A: We will prove the given statement.

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: If p and q are distinct primes and a is an integer such that aº = a(modq) and aª = a(modp), then

A: This is a problem of theory of congruence, number theory.

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

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Q: (d) For all integers a and b, if ab = 7 (mod 12), then either a = 1 (mod 12) or a = 7 (mod 12).

A: (d) For all integers a and b, if ab=7 mod 12, then either a =1 mod 12 or a =7 mod 12

Q: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

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: 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: 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: Write a formal proof for the following. For all integers a and b, if a ≡ 1 (mod 4) and b ≡ 3 (mod…

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Q: 12. Prove that if gcd(a, n) b, then ax = b (mod n) has no solutions.

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Q: Let a E Z. Prove that if a = 2 (mod 6), then a? = 4 (mod 12).

A: We will prove this by basic result of congruence

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: If a and b are integers such that a? = b² (mod p), where p is prime integer, then a) a = b (mod p)…

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Q: 2. Prove that if a, b, c, d, and m are integers such that m > 0, a = b mod m and c= d mod m, then…

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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: Prove that whenever x ∈Zand d ∈Z+, if d does not divide (x2 −3x), then x mod d does not divide 3

A: The given statement is False.  Consider, x = 5 and d = 4. Then we see, x2 - 3x = 52 - 3x5…

Q: Prove that for any integer a, at = 0 (mod 5) or aª = 1 (mod 5)

A: To prove for any integer a,  a4≡0mod5 or a4≡1mod5.

Statements and Logic

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