p is a prime number; a is an integer; Let (a, p) = 1 and the exponent of a with respect to mod p is 3 . In this case, show that the exponent of the integer 1 + a with respect to mod p is 6.

Q: 9. Consider modulo 11. a. Find 35 mod(11). b. Find a negative integer x such that x mod(11) is…

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

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Q: 3400 m is prime numbers, then compute 27 (mod m) 2. If m is natural numbers, show that a (mod m ) =…

A: Modular arithmetic: a≡b mod c means c divides a-b i.e., c|(a-b). So there exists a constant k such…

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

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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: Let x be an integer coprime to 1450 that is not congruent to 1 modulo 1450. Suppose we know that x1…

A: The given two MCQs are exactly same. Hence we will just solve the first one. And we will use the…

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: an is an even number, show that a2 = 0 (mod 4), and if an is an odd number, show that a2 = 1 (mod…

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Q: 1.Let a, b, and c be integers, where a≠0. If a | b and a | c, then a | (b + c)            True or…

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

Q: Take p and q are primes and p=3mod4 and q=3mod4. Take n=pq such that an integer z has a square root…

A: Given that р and q are primes and p=3mod4 and q=3mod4.

Q: Let p be a odd prime number and b, c € Z. Assume that there exists l e Z, such that l² = b² – 4c mod…

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

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Q: Prove or disprove If ac = bc(mod m), then a = b(mod m). If n| m and a = b(mod m), then a = b(mod n).…

A: Hello. Since you have posted multiple questions and not specified which question needs to be solved,…

Q: a) Let p be an odd prime. Show that a = -1 mod p has a solution if and only if p = 1 mod 8.

A: Consider the provided question,Hello. Since your question has multiple parts, we will solve first…

Q: Consider p> 2 a prime number, justify that: (P,') = (-1)" (mod p) to any 0 sn sp- 1,n EN. 1.1 1.2…

A: In the given problem we use the definition of combination. Then the problem is solved by…

Q: Let p be prime. Suppose a, b ∈ ℤ and ab congruent to 0 (mod p). Prove that either a congruent to 0…

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Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

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Q: Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1.…

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Q: Prove or disprove If ac = bc(mod m), then a = b(mod m). If n | m and a = b(mod m), then a = b(mod…

A: According to the given information, it is required to prove or disprove the given statement:

Q: 15. Give examples of the following. Justify your answers.: (b) A set X such that |P(X)| = 8. (c)…

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Q: Identify if the given statement is true or false: I. If a is an integer, the (a^2 is congruent to 0…

A: Given some statements. Check whether these statements are true or false.

Q: Let p = 2^k + 1 be a prime for some k ≥ 1. Let a be any integer such that gcd(a, p) = 1 and a is NOT…

A: Consider the given information:

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

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Q: - Show that (a + b)" = a + bn mod 2 for all integers a, b, and n with n > 0. Hint: Try looking at…

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Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: Prove that for any prime, p = 1 (mod 4) there are exactly (p-1)/4 incongruent values of a 0 (mod p)…

A: By theorem (a,p)=1 then xn≡a modp has g.c.d(n,p-1) solution or no solution according as…

Q: True or False For arbitrary integers a and b, a ≡ b(mod n) if and only if a and b do not have the…

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Q: 20. Are the following statements true or false? Either prove the statement is true or provide a…

A: PART A For the integers a and b, a.b=0(mod 6) This implies that either a=0(mod6) or b=0(mod6). So,…

Q: Let p be prime. (a) If a 1 (mod p), show that 1+ a + ...+ aP-1 = 1 (mod p). (b) If a -1 (mod p),…

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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: Consider modulo 11. a. Find 35 mod(11). b. Find a negative integer x such that x mod(11) is…

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Q: Suppose that each of a, b, and c is an integer and m is a positive integer. Show that if a = b(mod…

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Q: m≥ 2 is an integer; a and b are integers each; (a, m) = (b, m) = 1; Let h be the exponent of a…

A: Given, m≥2 is an integer, a and b is an integer; a,m=b,m=1 and, exponent of a = h according to…

Q: If p and q are distinct primes and a is an integer such that aº = a(modq) and aª = a(modp), then

A: This is a problem of theory of congruence, number theory.

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: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

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: Give an example to show that a ^2 ≡ b^2 (mod m) need not imply that a≡ b (mod m)

A: Here I listed given example of above statement.

Q: Prove the following:   a. If n is an integer and n + 3 is even, then n is odd by using a proof by…

A: Remark: Proof by contrapositive is based on the fact that an implication is equivalent to its…

Q: Let p be prime. Suppose ab≡0 mod p and a ≢ 0 mod p. Apply Euclid’s Lemma to prove b ≡0 mod p

A: Given that, Let p be prime. Suppose, ab≡0 mod p and a≢0 mod p. To prove: b≡0 mod p…

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: Using Wilson's theorem, to show that if p is a prime and p = 3 (mod 4), then ()! = +1 (mod p).

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

A: Fermat's theorem

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: 5. Show that (a + b)n = a + bn mod 2 for all integers a, b, and n with n > 0. Hint: Try looking at…

A: Given data: (a-b)n= an +bn mod 2 for all intergers a,b and n with n>0.

Statements and Logic

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