Find all integers X satisfying X ≡ 1 mod 6, X ≡ 3 mod 8, X ≡ 7 mod 9

Q: Let n be a positive integer. Prove 41+3n mod 9.

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Q: Prove that if ca = cb (mod n), then a = b (mod n/d) , where d = gcd(c,n). %3D

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Q: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

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Q: Assume that k mod 15 4. Compute 11k mod 15. Show all your work clearly.

A: Topic:- basic maths

Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: Express each nonnegative integer a less than 15 as a pair (a mod 3, a mod 5).

A: The values of a could be, 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14.

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: 7. Find a whole number x in the interval [0, 30] that satisfies the following x = 1 mod 2 x = 2 mod…

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Q: 12. Prove that if gcd(a, n) { b, then ax = b (mod n) has no solutions.

A: To prove the given statement, it is equivalent to prove its contrapositive statement which is "if…

Q: (a) Find three distinct integers satisfying the equation r = 1 mod 11. (b) Prove that if k is an…

A: a≡bmod m means a-b is divisible by m where a and b are integers. For example 13≡3 mod 5⇒13 - 3 is…

Q: 14. Find all 2<x<19999 such that x=-x(mod 20000) Ans:

A: To find all x where 2≤x≤19999 such that x2≡-xmod 20000.

Q: Find all integers a such that = 6 mod 29. a. 21 (mod 29) O b. 1,0 (mod 29) O c. 10, 21 (mod 29) O d.…

A: According to Fermat's little theorem if p is any prime and a is any integer, then ap-1≡1modp here p…

Q: Prove the statement by contradiction.

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Q: Find all solutions in {0, 1,2, ..., 12} to x26 + 15x13 = 12 mod 13.

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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 positive integers a is the congruence ax4 ≡ 2 (mod 13) solvable?

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Q: Let n be a fixed positive integer greater than 1. If a mod n = a' andb mod n = b', prove that (a +…

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Q: 13. a. Prove that for every integer n 1, 10" = (-1)" (mod 11). |

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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: Number theory

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Q: 13. a. Prove that for every integer n > 1, 10" = (-1)" (mod 11).

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

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Q: Find all x EZ221 such that x2 =97 (mod 221).

A: To find : all x∈ℤ221 such that x2≡97mod 221 Pre-requisite : Properties of Legendre symbol - 1. ap=1,…

Q: Prove that for every integer n, n3 mod 6 = n mod 6.

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Q: Show that 18! ≡ -1 (mod 437).

A: We have to show that: 18!≡-1mod437

Q: Let m1, m2,…, mn be pairwise relatively prime integers greater than or equal to 2. Show that if a ≡…

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Q: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡−1 mod 7

A: We will prove the given statement.

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: mod n, then x ≡ 0 mod n or

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Q: Suppose that a1 ≡ b1 mod (m) and a2 ≡ b2 mod (m) -Show that a k 1 ≡ b k 1 mod (m) (hint: use…

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Q: Suppose that a and b are integers, c b = 3 (mod 19). Find the integer c that a) с%3D 13а (mod 19).…

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Q: Use Fermat's theorem, show that if ged(a, 155) 1 then a1 = a (mod 15

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Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

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Q: 27 = x(mod 5) is true if x =

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Q: Prove that (a mod 2)(b mod 2) = (ab) mod 2. ANY METHOD of proof.

A: Prove that a mod 2b mod 2=ab mod 2

Q: Establish that if a is an odd integer, then 2" a“ = 1 (mod 2" *2) for any n 2 1.

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Q: 12. Prove that if gcd(a, n) b, then ax = b (mod n) has no solutions.

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Q: Prove that if a = b mod n and c= d mod n, then ac bd mod n.

A: Given if a≡bmodn and c≡dmodn We have to show that ac≡bdmodn. Now, a≡bmodn…

Q: Prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)

A: We have to prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)

Q: Find all solutions to the congruence 10x≡5 mod 15 (b) Prove that 7 divides 32n+1+ 2n+2for any…

A: In  (a) we use the extended Euclidean algorithm to solve the problem. In (b) we use the mathematical…

Q: Prove: for any integer x, if x mod 3 is not 0, then (x2−3x+ 1) mod 3 = 2.

A: To prove :     for any integer x, if xmod 3≢0, then x2-3x+1mod 3=2   Pre-requisite : 1. a≡bmod n…

Q: 3. Find all integer solutions to the pair of congruences (if any) x = 1 (mod 2) x = 2 (mod3) x = 3…

A: To Find: Integer solutions to the pair of congruences(if any) x≡1(mod 2), x≡2(mod 3), x≡3(mod 5)…

Q: The least positive integer x satisfying 22010 = 3x (mod 5) is

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

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Q: 2x = 1(mod 5) 31( mod 7) 5x 9(mod l)

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