Suppose that a1 ≡ b1 mod (m) and a2 ≡ b2 mod (m) -Show that a k 1 ≡ b k 1 mod (m) (hint: use induction)

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

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Q: Prove that if ca = cb (mod n), then a = b (mod n/d) , where d = gcd(c,n). %3D

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Q: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

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Q: 5. Prove that m(n) + nº(m) = 1 (mod mn) if (m, n) = 1. %3D

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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: Show that if p is a prime number then every equation of the form ax ≡ b mod p with a 6≡ 0 mod p has…

A: Given p is a prime number and 6≡0 mod p. Therefore either p=2 or p=3. Let d=gcda,p. We know that if…

Q: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

Q: If p is a prime with p=3 mod 4 and a is quadratic residue mod p, then -a is a quadratic residue mod…

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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: 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: 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: [Hint: Proceed by induction on n.] 16. Use the theory of congruences to verify that 89| 24 – 1 and…

A: The solution is given as follows :

Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

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

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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…

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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 m)…

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

Q: Exercise 16. Prove (without induction) that B" = 0 mod B for all positive integers B and n. Hint:…

A: Congruence and proof without induction

Q: Number theory

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Q: Find the least positive integer n that satisfies the congruence 7 ^128 congruent to n (mod13)

A: Given, 7128≡nmod 13 Find the value of n that satisfies the above congruence.

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: 4 If a = b (mod n), then a" = b™ (mod n) for every positive integer m.

A: It is given that  a≡b (mod n), then am≡bm (mod n) for every positive integer m. It will prove by…

Q: Suppose that m is a positive integer. Use mathematical induction to prove that if a and b are…

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Q: Consider the congruence mod 6 for integers. Z6={[0], [1], [2], [3], [4], [5]}. Prove the product of…

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Q: 1. Suppose m, nEz* are relatively prime. Prove that for all a, bEZ, a =b (mod mn) if a =b (mod m)…

A: Note that: a is congruent to b modulo m if and only if m divides (a-b).

Q: Let m1, m2,…, mn be pairwise relatively prime integers greater than or equal to 2. Show that if a ≡…

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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: 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: Show that y = x (mod p) leads to -y = -x (mod p).

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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: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

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Q: (b) Prove that multiplication mod n is commutative.

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Q: b) Prove that if n2 (mod 3) = 1 then n is not divisible by 3.

A: The required proof is as follows -

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

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Q: Find a such that 5= 7* modulo 13.

A: Given: 5≡7xmodulo13. The objective is to find the value of x.

Q: Suppose that a, b, m, and n are integers (n, m positive) and suppose we know a = b (mod n) and n | m…

A: We first show how the conclusion is true from that we choose the right option.

Q: Prove that (a mod 2)(b mod 2) = (ab) mod 2. ANY METHOD of proof.

A: Prove that a mod 2b mod 2=ab mod 2

Q: If a procedure can be broken down into n tasks, where each I task can be performed ti Prove via…

A: To prove that if a procedure can be broken down into n tasks, where each i task can be performed ti…

Q: Show that if p is an odd prime and a is an integer not divisible by p, then the congruence x2 ≡ a…

A: According to the given information, Let p is an odd prime and a is an integer not divisible by p…

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: Use Wilson's Theorem to prove that for primes p congruent to 1 modulo 4 the quadratic congruence 1'…

A: Given The quadratic congruence x2≡-1(mod p) To prove  x2≡-1(mod p) has a solution if p≡1(mod 4)…

Q: Use induction to prove for every positive integer, 4 x 10n = 4 or 7 (mod 33)

A: Suppose P(n) be any mathematical statement in terms of n. Then Principle of mathematical induction…

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: If a = b(mod n²) then a = b(mod n)

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Q: Solve for d using Euclid's Extended Algorithm 3d ≡ 1 mod 5200

A: Given 3d≡1 mod 5200 . The objective is to solve for d using Euclid's Extended Algorithm.

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…

Q: Show that if p is a prime number then xy ≡ 0 mod p implies that x ≡ 0 mod p or y ≡ 0 mod p

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