(12) Let a,b, m, nE Z. Assune if greater than ave Prove mln and a = b (mod n) that then a = b (mod m) 4 4

Q: 2-) Find all in tegers x such thet 5x = 12 (mod 7) 10x = 4 (mod 1) 7x E 16 (mod 10d ond 1occ < x <…

A: Given: 5x≡12(mod 7) 10x≡4(mod11) 7x≡16(mod 10) and 1000≤x≤3000

Q: Problem MCD = Greatest common divisor (GCD) t them.a, b, c and m €Zsuch that m 22. a-). Prove that a…

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Q: 9. Let a and b be integers. Prove that if a =7 (mod 8) and b = 3 (mod 8), then: (a) a + b = 2 (mod…

A: Your questions solution below in images

Q: For each statement, indicate whether it is true or false. Then prove or disprove it. (a) For all SC…

A: (a) FALSE.  Consider S ⊆ℤ, defined by  S = { 2+8k | k ∈ℤ } U { 2+12r | r∈ℤ }.  Then the given…

Q: (c) For each integer a, if a 4 (mod 8), then a 2 (mod 8).

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Q: 1. Let a, b e Z. Prove that if a = 0 (mod 5) and b = 2 (mod 5), then a² + b² = 4 (mod 5).

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Q: a) Prove that 10" =+1 (mod 11) for any n EN. b) Suppose the integer r has digits rI-1I, I0. Prove…

A: Want to prove:(a) 10n≡±1 (mod 11) for any n∈N(b) If the integer x has digits xnxn-1.....x1x0 thenx≡0…

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: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

Q: 4. Let a, b, c, d, n € Z such that a = b (mod n) and c= d (mod n). (a) Prove that a+c=b+d (mod n).…

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Q: Q6. Using the definition of x = y mod N, justify the rule x = x' mod N, y = y' mod N = x x x' = y x…

A: Using the definition of modulo N.

Q: Let a = (1 73)(5 49 2) and B = (2 3)(7 4)(5 18) be elements of S10. Then, a B is O (12974)(3 58) O…

A: Let α=1735492 and β=2374518 be elements of S10, then α2β-1 is

Q: Let a, b, c and meZ, and m 2 2 1. Prove: if a =b (mod m), then GCF (a, m) = GCF (b, m) %3D m 2.…

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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: 5. Let p = 61. Find all incongruent elements modulo 61, that have order: (а) 5 (b) 10 (с) 60

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Q: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

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Q: Exercise 16. Prove (without induction) that B" = 0 mod B for all positive integers B and n. Hint:…

A: Congruence and proof without induction

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: Let n be a positive integer and let ged(b, n) = 1. Show that bx by (mod n) if and only if x=y(mod…

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Q: 5. The Fibonacci numbers are defined as: fo = 0 , fi = 1 , and fn = fn-1 + fn-2 , for n > 2. Give a…

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Q: Compute the order of 2 with respect to the prime moduli 3, 5, 7, 11, 13, 17, and 19.

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Q: 7. Let p be a prime number and n= 2P – 1. Prove that 2"-1 = 1 (mod n).

A: Let p be a prime number and n=2p-1. Prove that 2n-1≡1 (mod n)

Q: IfOcy (mod n) and e> 0 then a) *= y (mod cn) b) x= y (mod n) ) x=y (mod) d) None of these :Select…

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Q: How many positive integers n less than or equal to 11,347 satisfy d(n) 1 (mod 2)?

A: We need to find a number of positive integers n≤11347 such that d(n)≡1(mod 2) i.e., , k∈Z.So, we…

Q: *1. What is the least residue of 56 (mod 7) 58 (mod 7) 19458 (mod 7)?

A: Note that : Since you have posted multiple questions, we will provide the solution only to the first…

Q: 2. Let A = {n? +n € Z:n e Z} and B = {r e Z : r³ – 3x³ + 2x = %3D O mod 5}. Prove or disprove: A C…

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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: compute: 11 31*

A: Let ⊕:Z37⊕ + Z37⊕  →Z37⊕ be defined as a⊕b=(a+b)mod 37

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: Find the integer n that has the following properties: i. gcd(n, 10) = 1 ii. iii. 9 = n38 (mod 10) -9…

A: Concept: The smallest positive integer d that can be a solution or that can be represented as a…

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

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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: Prove directly if the statements that are true, give counterexamples to disprove those that are…

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

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Q: 20.What is the value of n such that (24  is congruent to 58 mod n) ? a.13 b.17 c.-34 d.-13

A: # as per the guidelines we are entitled to solve one question at a time, please resubmit the other…

Q: Let min EN 7 atR he such that ca,m)=1, Ca,n)=l ģ (min) =L, Let K= Om(a) } j=OnCa). Suppore Prove…

A: Let m, n∈ℕ and a∈ℤ be such that a, m=1, a, n=1 and m, n=1. Let k=oma and j=ona. Suppose j | k, prove…

Q: Let n1, n2, n3 be three numbers that are pairwise relatively prime so that (n1, n2) = (n2, n3) =…

A: Given that n1,n2 and n3 are three numbers that are pairwise relatively prime. We have to prove…

Q: 10 =1 (mod 9) Select one: O True O False

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Q: Let x, y, u, v € Z and d E N. Prove that if x =y (mod d) and u = v(mod d), then xu = yv (mod d).

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Q: 25 Prove that mod 7 5 for all Gindegars and bin b mod 7-6

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Q: 6.) Prove by induction that 1/5 + 1/3/53 13 + *** ㅗ 402-1 n for every nz | 2n+1

A: