For which primes p > 7 is 7 a square mod p? Your answer should be of the form p = {a1, a2,... , a} mod n, where a; E {1, .,n - 1} for all i= 1,..., k. mod

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

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Q: For inputs of 0 and 1 and mod 2 arithmetic describe the output ofthe formula z + xy + xz in the form…

A: As per the question we have to describe the output of the formula : z + xy + xz For the inputs of 0…

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: 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: Let n = 17. Sha

A: Given: Value of a, b, n. We have to check whether a≡b mod n    or   a≢b mod n.

Q: Congruence conditions (mod 20) for primes p for which -5 is a square (mod p).

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Q: He have to Show that lon = (-1)" (mod 1") shall use mathematical moluction Let Pin) := lon a (47"…

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Q: For which odd primes p is 5 a square modulo p?

A: As we know that  a is Quadratic residue mod p if and only if   a(p-1) /2 =1 mod p . If p and q are…

Q: III. Fill-in the blanks. Tare more than 1 answer. Choose any. 1. 46 = 26 mod 2. 156 = mod 15 3. 41 =…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve the first three…

Q: Which one of the following is correct? a. 30 쓰 3 (mod 5) O b. 25 3 (mod 4) O c. 31 3 (mod 7) Od 20 3…

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Q: If x = 2 (mod 5) and y = 4 (mod 5) then 2x – 3y =? (mod 5)

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Q: Congruence conditions (mod 20) for primes p for which -5 is a square (mod p).

A: The proof is based on a simple application of the law of quadratic reciprocity and the…

Q: If a = b (mod n) and b =c (mod n) then which of the following is false. a = c (mod n) a = c3(mod n)…

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Q: Find the smallest positive integer for which x mod 3 = 2 and a mod 4 = 3 What is the next smallest…

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Q: 3. 78 + x = 3 ( mod 5) 4. 89 = (x + 3) ( mod 4)

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Q: a.) Determine if x< = 2 (mod 7817) has solutions. b.) Determine if x< = 3760 (mod 4861) has…

A: x2≡2(mod 7817) has a solution if and only if  2 is a quadratic residue of 7817. Euler's criterion…

Q: Using Euler's Theorem, simplify the following numbers: o = 3, q = 5, x = 2, and a = 2. Which of the…

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Q: If n| m and a = b(mod m), then a = b(mod n).

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Q: y²=x³-5x+3 (mod5). solve for all x and y, when x is at least 0, and y is at most 4

A: Using congruence and modular arithmetic we solve the given equation. We substitute the value of x…

Q: Which of the following are true modulo 12? (1) Does 5z = 6 mod 12 have a solution? How many? (2)…

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

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Q: Hanb(mod n) and b c(mod n) then which of the following is trae. (a) anc (mod n) () 2=c(mod n) (c)…

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Q: Does 5x = 6 mod 12 have a solution? How many?

A: As per our guidelines we are supposed to answer only One question  if there are multiple questions…

Q: QUESTION 2 Which is correct? 72+ 5 = 32 (mod 9) 74 = 164 (mod 10) 19× 27 = 7– 3 (mod 6) 24 – 27 = 5…

A: 72 + 5 = 49+5             = 54             = 9 (mod 9)             = 32 (mod 9) Hence 1 is true. 7 4…

Q: Give all non-negative solutions for 10r =-17 mod 4.

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Q: Find an integer x that satisfies the equation. 5* = 4(mod 11) 17x?=10(mod 29) i. %3D ii.

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Q: 5. Which of the following are inverses of each other? (a)3, 7 w.r.t. mod 20 (b)2, 5 w.r.t. mod 9…

A: Explanation of the answer is as follows

Q: In the following question: " Let a, b ∈ Z. Show that if a ≡ 5 (mod 6) and b ≡ 3 (mod 4), then 4a +…

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Q: Which one of the following is correct? a. 30 3 (mod 5) O b. 25 3 (mod 4) Oc. 31 3 (mod 7) O d. 20 3…

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Q: 35. Let p be an odd prime number. Prove the following statements. (a) (7) = 1 if and only if p = 1,…

A: We have to prove given statement if p is an odd number.

Q: 1. Show that if pla + a + a? + a+1 and a -1 mod 5 then P=1 mod 5. Hint: a -1= (a - 1)(a* +a + a? +…

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Q: 4. Consider the following statement: For all n N and x = Z, x² = 1 mod n Which of "if", "only if",…

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Q: The natural numbers are N= {1,2,3,..}. The prime numbers are P = {2,3,5,7...}. True or false? ØSN…

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Q: Find the integer a such that a. a = -43 (mod23) and -22 ≤ a ≤ 0.

A: Find the integer  such that  and .

Statements and Logic

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