Prove that if n is an odd Integer, then n² = 1 (mod 8) 2

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 that for all integers n ≥ 1, 10" = (-1)" (mod 11). Use part (a) to prove that a positive…

A: (a) Let n≥1 be an arbitrary integer. We know that 10≡10mod 11≡-1mod 11. Now consider 10n:…

Q: Find the smallest positive integer k such that 3k = 1 (mod 19).

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Q: Show that if p is an odd prime, then 2(p - 3)!= -1 (mod p) Prove that if n has an odd number of…

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Q: Find an integer n such that n = 1 mod 7, n = 3 mod 8, and n = 2 mod 9.

A: We can solve this system of congruences using the Chinese Remainder Theorem.

Q: Prove that : If p is a prime number and p does not divide a then a- 1(modp).

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Q: Suppose that 2n−1 ≡ 1 (mod n). Is n necessarily prime?

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Q: that there does not exist an integer n such that n= 21),

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Q: The notation x = y mod n is used to indicate the unique integer x in the range 0 :$ x < n such…

A: Given statement is that the notation x = y mod n is used to indicate the unique integer x in the…

Q: 2 Number theory a) For any integer n, show that n² = 0 (mod 4) or n² = 1 (mod 4). b) Use the result…

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Q: 4. For any integer n, show that if n³ = 1 (mod 47), then n = 1 (mod 47).

A: To show that if n3≡1(mod47), then n≡1(mod47), it can use proof by contradiction.Assume that…

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: 3. Prove: for every integer n and for every odd integer a² = 1 (mod 2n+2)

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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: 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: Prove by contraposition that for all integers n, if n is odd, then 5n 4 is odd.

A: We need to prove if n is odd, then 5n - 4 is odd by the method of contraposition.  The…

Q: (iii) Lagrange mean value theorem. (iv) Cauchy mean value theorem.

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Q: *discrete math* Prove that for all integers a, b, if a and b are congruent to 1, mod 3, then their…

A: An integer n is congruent to 1, mod 3, if and only if there is an integer k such that n = 3k + 1.

Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: 11. Prove that a¹ = 1 (mod 5) for every integer a that is not divisible by 5.

A: We have to prove  for every integer  that is not divisible by

Q: Use a Proof by Contradiction to prove that there is no integer a such that a congruent 5 (mod 14)…

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Q: Let p be prime and a an integer not divisible by p. Prove that if a²"-1 (mod p), then a has order…

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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: Use induction to prove that for every integer 15"+2 + 162n+1 is divisible by 241. n > 0,

A: To use mathematical induction to prove the given statement

Q: Let a be an integer. Prove the following statements: (a) If a is even, then a 2 ≡ 0 (mod 4). (b)…

A: We have to prove given results:

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: DuppOse that a anа т аге intege Such that a, аre al diпerent IMod- ulo m, with am-1 = 1 mod m. Show…

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

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Q: Najabhai

A: The objective of the question is to find all integers n that satisfy the equation n² + 503 = 0 (mod…

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

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Q: 16. Prove that an integer (a,-1an-2 · . . ao)10 is divisible by 11 if and only if ao + a2 + a4 + . .…

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

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Q: he smallest positive integer x such that x = 22 (mod 25), x =

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Q: 10. Prove that a³ =a (mod 6) for every integer a.

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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 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: If a is an integer such that: 2 ła, 5ła Show that (a)⁴ⁿ = 1(mod 40) for any positive integer n

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Q: a) Let a be an integer with a > 0. The integer a is even if and only if a = 0 mod 2. Note to self:…

A: (a) Let a be an integer with a &gt; 0.       We need to prove that a is even if and only if a ≡ 0…

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.