For which positive values of k is nk = n (mod 7) for all integers n?

Q: For which odd primes p is 3 a square mod p?

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Q: if b is modulo m and a is another unit with a=b mod m, we say that d is the---------of a modulo m to…

A: As with such a large amount of concepts we are going to see, congruence is straightforward, perhaps…

Q: The integer n = 12700140x3243243 is missing the digit x. a) If n congruent to  3 (mod 9), then what…

A: According to the given information, Consider the given integer n:

Q: Let x be an integer. Prove that, if x mod 9 = 5, then 9|(x^2-2x+3)

A: Given: x mod 9=5 then, 9|x2-2x+3|

Q: Fermat's "Little" Theorem states that whenver n is prime and a is an integer, a"-l = 1 = 1 mod n a)…

A: Given a =463 and  n=953 We need to find x such that                   463952 ≡ x (mod 953).

Q: 2. Give an example to show that a? = b2 (mod n) need not imply that a =b (mod n). 3. If

A: Given below the detailed solution

Q: Use the table in VII to give a formal proof of the following. For all integers a and b, if a = 1…

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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: For which odd primes p is 5 a square mod p?

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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: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

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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: Let n be an integer greater than 1. If (n-2)! ≡ 1 mod n, then Prove n is a prime number.

A: Use the Wilson theorem and different case to solve this problem as in next step.

Q: Suppose r is a primitive root mod a prime p such that p = 3 mod 4. Prove ord,(-r) =

A: Given: p≡3mod 4 To find: ordpp-g=p-12

Q: a. Let m be an integer such that m mod 5 = 2, what is 8m mod 5? b. Let n be an integer such that n…

A: Consider the given information: Let m be an integer such that m mod5 = 2. Then m=5k+2   where…

Q: 2

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Q: 2. Give an example to show that a (mod n). b (mod n) need not imply that a =b

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Q: Suppose that p is a prime number. Given this information prove that if x is an integer such that x^2…

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Q: Given any odd prime p, prove that: 2(p−3)! ≡−1 mod p. (Hint: Start with 2(p−3)!≡x mod p. Then work…

A: Let p be any odd prime number.  So we start with  2p-3!≡x mod p Also we have p-2≡-2 mod pp-1≡-1 mod…

Q: The last two digits of 52044 is 44. If ab = 0 (mod 3), then a = 0 (mod 3) or b = 0 (mod 3). F 10. F…

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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 2 is a primitive root mod 3^e for all e>=1

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Q: For all integers a and b, if a = 7 ( mod 8)and b = 3 ( mod 8), then (a + b) = 2( mod 8).

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Q: Prove the following results. (I). If n = 2(mod 4), then n is the sum of three squares of integers.…

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Q: If a has order n-1 mod n, then n is prime. O True O False

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

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Q: Suppose that a and b are integers, a = 4(mod 13), and b = 9(mod 13). Find the integer c with 0<c< 12…

A: We will use the basic properties of congruence modulo n relation to answer this question.

Q: The least residue of 5500 mod(17) is Select one: O none O 7 O 13 O10 O 8

A: The least residue is, 13 The detailed solution is as follows below :

Q: Let p > 3 be a prime. Show that if p2+ p + 1 is also prime, then p =5 mod 6.

A: We prove the result by the method of contradiction.

Q: (d) For all integers a and b, if ab = 7 (mod 12), then either a = 1 (mod 12) or a = 7 (mod 12).

A: (d) For all integers a and b, if ab=7 mod 12, then either a =1 mod 12 or a =7 mod 12

Q: The order of 5 mod 17 is

A: “Since you have asked multiple question, we will solve the first question for you. If you  want any…

Q: The order of 5 mod 17 is *

A: Since ϕ(17) = 16 Therefore order can be the divisor of 16 So the order can be 1,2,4,8 or 16

Q: Consider numbers of the form 9" mod 13, where n E Z+. We claim that all of these numbers will only…

A: First principle of mathematical induction : Suppose Pn is a statement where n∈ℤ+ . The statement is…

Q: Let n be an integer. Prove that, if n mod6 = 2, then n2mod12=4

A: We apply concept of modulus here to solve this.

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.

Q: 29. How many different values are there for 4n(mod 7), as n ranges over all positive integers? 30.…

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Q: In(n² = n(mod 10)), domain is the set of positive integers.

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Q: Prove that if n is an odd integer, then n^2 is congruent to 1 modulo 8.

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Q: Write a formal proof for the following. For all integers a and b, if a ≡ 1 (mod 4) and b ≡ 3 (mod…

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Q: Let p >2 be a prime. Prove that [(p-2) = (-1)/2 (mod p).

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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: Let a, b, and c be integers and n a natural number. If a ≡ b mod n then a+c ≡ b+c mod n. How do I…

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Q: Q2/let n be a fixed positive integer and a,b be arbitrary integers. Then a = b (modn) iffa and b…

A: We have to prove that Given statement

Q: For n=195, by Fermat's theorem d-2 a) a (modn) b) 1(modn) c) 0(mod n) d) a(mod n)

A: Given:                 n=195 To find:                an-2=? by using Fermat's theorem Fermat's…

Q: If a and b are integers and n is a positive integer, prove that a mod n =b mod n if and only if n…

A: If a and b are integers and n is a positive integer,    Formula used: a≡b mod n if and only if  a-b…

Q: Show that if a is an even integer, then a =0 (mod 4), and if a is an odd integer, then a =1 (mod 4).

A: Suppose a is an even integer Then a=2m for some integer m a2=(2m)2 a2=4m2Clearly 4|4so 4|4m2so…

Q: Prove or disprove: If a = b(mod n²) then a = b(mod n)

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Q: Prove the above property. That is, prove that if a mod n = k and b mod n = j, then (a + b) mod n =…

A: If a mod n =k and b mod n=j show that a+b mod n =j+k mod n. Definition: If x mod y=z, then y | x-z.…

Q: For any integers m and n, if m mod 11 is 6 and n mod 11 is 7, what is m+n mod 11? 13 9 42

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Statements and Logic

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