6. Let m be an odd positive integer. Prove that every integer is congru- ent modulo m to one of the even integers 0, 2, 4, 6, ..., 2m-2.

Q: 8. Let m 2q + 1 be an odd positive integer. Prove that every integer = is congruent modulo m to a…

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Q: if a equiv b (mod n) , then a⁴ equiv b⁴ (mod n) for any positive integer 4.

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Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: Express each nonnegative integer a less than 15 as a pair (a mod 3, a mod 5).

A: The values of a could be, 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: 1. Find all solutions (a) 2x + 3 = 7 (b) 2x + 3 = 7 to the following congruences.. (mod 9) (mod 20)

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Q: Let a and b be integers satisfying a=3(mod 11) and b=8 (mod 11). If c=3a+4b then None of these c=9…

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Q: 1) Compute the order of a modulo 7 for a = 1,2,....6. Find all primitive roots modulo 7.

A: Order of the element

Q: . Determine whether or not x³ + 2x + 1 and x4 + 3x - 2 are congruent mod (x² + 2x + 2) in Q[x].

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Q: Prove that 2 is not a primitive root modulo 31.

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

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Q: Give a definition of an inverse of a number a modulo m.

A: if the two integer a and m modulo then  let x is an integer value so the x is inverse of an only if…

Q: Prove that the order of 3 modulo 19 is a divisor of 18.

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Q: Find all solutions in {0, 1,2, ..., 12} to x26 + 15x13 = 12 mod 13.

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Q: Test2

A: We know that2198=2128×264×24×22Also we have22≡4(mod 397)24≡16(mod 397)28≡256(mod…

Q: (a) Prove that 2 and 3. are both primitive roots modulo 5.

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Q: (7) (Z/50Z)*. Which of the other elements of (Z/50Z)* are also primitve roots modulo 50? Find a…

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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: Suppose that a is an odd integer and (a, 91) = 1. Prove that a12 ≡ 1 (mod 1456).

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Q: Find all the primitive roots modulo 125.

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Q: (2) Compute the order of a modulo 11 for a = 1,2,..., 10. Find all primitive roots modulo 11.

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Q: a-b|m mla-b O m|b-a O both mla-b and m|b-a O

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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 m 2q be an even positive integer. Prove that every integer is congruent modulo m to a unique…

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Q: 8) Prove that 2 is a primitive root modulo 53.

A: We have to Prove that 2 is a primitive root modulo 53.

Q: (c) (n² + n+ 41)² = 1 mod 12 , for every positive integer n

A: We know that,

Q: Is 3 a primitive root of 43? Use your answer to determine all positive integers less than 43 whose…

A: To determine whether 3 is a primitive root of 43, we need to check if it generates all the positive…

Q: Let (a,b)=1 with a = c (mod b) and bª = c (mod a) for some integer c. Prove that ab+(b). +ba+(a) = c…

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Q: 10. Let m and n are two positive integers with m > n. Let a be a positive integer. Prove that a2" +1…

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Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

Q: Show that there are no integers a, b such that a² + 2ab+26² = 100000000003. Hint. Use mod 4…

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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: 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: Let a and b be relatively prime positive integers. Show that a(b) + b(a) = 1(mod ab).

A: Solution:

Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

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

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

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Q: 5. Prove that every integer is congruent modulo 9 to one of the even integers 0, 2, 4, 6, ..., 16.

A: To prove that every integer is congruent modulo 9 to one of the even integers

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

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Q: Show that 18! ≡ -1 (mod 437).

A: We have to show that: 18!≡-1mod437

Q: Find oll integers Such thot 24! x =1 Cmod 2) 5x = 3. (mod 8) and 1500 < X <2000

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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: Suppose that p is a prime number and g is primitive root. Prove that the least residues of 4 9²,9ª,…

A: Step 1:To prove that the least residues of g2,g4,g6,gk,...,gn−1 are all quadratic residues modulo n,…

Q: Let 1 ≤k≤p-1 where p is a prime number. Then prove that (₁7¹) = (-1) (modp) k

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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: 4) Let n 2 2 be an integer, such that 2" – 1 is prime number (e.g. if n = 7 then 2" – 1 = 127 is a…

A: Solution