Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i< n), then aļɑ2 • · · An =1 (mod 3).

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: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

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: • n = 11, o(n) = and a =_ i a (mod n) j = ordn(a¹) is a primitive root. 1 2 3 4 5 6 7 8 9 10 List…

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

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Q: Show that if p is an odd prime and a and b are integers with a ≡ b (mod p), then

A: According to the given information, let p be an odd prime and a and b are integers with a = b…

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

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Q: Find n ∈ {1, 2, 3, . . . , 28} that satisfies 26! ≡ n (mod 29).

A: Given ,              26! ≡ n mod(29)  Then we have to find the  value of n∈  1 , 2 , 3 , .........,…

Q: 5. Find an integer x between 0 and 776 such that x = 13 (mod 21) x = 5 (mod 37)

A: Set N = 21 × 37 = 777. Then , m1  = N/21 = 777/21 = 37 and m2 = N/37 = 777/37 = 21 Now we need to…

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

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Q: 2. Find the integer a such that a = 2*(mod31) and -16 < a< 15. а) 16 b) -16 с) -15 d) 15 e) 8

A: Here we have to find a.

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

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Q: Prove that's is prime p, then (a + b)² = (a² + b²)(mod p), Va; b € Z

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Q: Find all integers between 1 and 20 that are zero divisors of mod 21

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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: Suppose that a and b are integers. a ≡ 7 (mod 13) and b ≡ 9 (mod 13). Find the integer c such that 0…

A: The given problem is to find the modular values of c for the given conditions with given integers a…

Q: a) If a| (21b−c) and 6c≅3b(mod2a), then a| 41c b) If a≅b(modc) and d|c then ?≅?(mod?) c) ∴(d|…

A: this is a question of modular arithmatic.

Q: 2

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Q: x

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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: let a,b,c,d, and n be integers with n>1. Prove that if a is congruent to b(mod n) and c is congruent…

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Q: III) Are there positive integers x and y suct that x³ = 2² +15? (Hint: for every integer x, x³ 0, 1,…

A: We shall show that no such positive integers exist.

Q: 7. Prove that every integer is congruent modulo 9 to a unique integer T such that -4≤r ≤ 4.

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Q: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

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: Prove by any method(contradiction/contrapositive or cases)   Prove that (a mod 2)(b mod 2)= (ab)…

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

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Q: 2) Find the smallest positive integer z, such that z = 13 mod 4 z = 20 mod 9 z = 11 mod 25 .

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

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Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

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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: "Proof 7: Suppose that a and b are both even. Then there exist integers k₁, k₂ such that a = 2k₁ and…

A: As per rule of bartleby we can solve only one question at a time and we can solve only three subpart…

Q: 6. Let m be an odd positive integer. Prove that every integer is congru- ent modulo m to one of the…

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