Prove that if a = b (modn) and m/n, then a = b (modm).

Q: Prove if a=b mod m then a^n=b^n mod m use Math induction

A: We have to prove that if a=b mod m then a^n=b^n mod m

Q: 3. Find integers x, y € [0,51] such that (a) 23x= 1(mod52) (b) 23y = 4(mod52)

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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: Let a, b be integers. What does it mean for a = b (mod 17)? Describe the elements of Z17. Compute…

A: a ≡ b (mod 17) means that a − b = 17k for some k ∈ ℤ. ∴  b − a = −17k = 17(−k); henceb ≡ a (mod…

Q: If a = b (mod m) and a = b (mod n) then a = b (mod mn)

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Q: Let m be a positive integer. Show that a mod m = b mod m if a ≡ b (mod m)

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Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: |Let m be a positive integer. Let a and b be integers such that a = b (modm). Use mathematical…

A: Solution

Q: Suppose a E Z. Prove that if a³ 1 (mod 5), then a # 1 (mod 5).

A: In this question, we use the method of contradiction to prove the desired result.

Q: Use Fermat's theorem, show that if ged(a, 155) = 1 then a121 = a (mod 155).

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Q: and cto (mod P) 37. Prove that if p is a prime and C#o (mod p), solutien then cx=b (mad p) has. a…

A: We need to  Prove that  p is a primeand  c ≢  0 (modp);then cx≡ b (mod p) has a unique solution

Q: Find the smallest positive integer x that solves the congruence: 7x≡5 (mod52)

A: 7x≡5 (mod52)---(1)

Q: Exercise 37 Find an n where x² = -1 (mod n) has a solution and an n where x² = –1 (mod n) has no…

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Q: 1. Prove that if ā and be Zn are invertible modulo n, so is ā - b. Is this the case for ā + b?…

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Q: Let n and a be positive integers and let d = gcd(a, n). Show that the equation ax mod n = 1 has a…

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Q: a) Show that ac = bc mod cd if and only if a = b mod d

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Q: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

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Q: 8, Suppose a,b € Z. Prove that a = (mod 5). b (mod 10) if and only if a = b (mod 2) and a = b

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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: Use Fermat's theorem, show that if gcd(a, 145) = 1 then a141 = a (mod 145).

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Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

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Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

Q: 3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6). (a) Check this…

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

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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 if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).

A: we will prove that gcd (a, n) = 1, which  means that a and n are  prime. Let m= gcd (a, n) = gcd (a…

Q: suppose a,b ∈ ℤ. Prove that a ≡ b (mod 10) if and only if a ≡ b (mod 2) and a ≡ b (mod 5).

A: In Solution we use divisibility formula that  If a and b are integers, a divides b if there is an…

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: 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 ifa≡b(modn)andb≡c(modn), thena≡c(modn). Note that you must use thedivides definition of…

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Q: (b) Prove that if a = 7 mod 10 and b = 6 mod 15, then a + b = 3 mod 5.

A: Our objective is to prove . Since you have posted multiple questions I am solving one.

Q: Prove that min (a) If a = b (mod n) and mln ,then a = b (mod m) (b) If a = b (mod n) and c =d (mod…

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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: Find an integer x satisfying the two congruences: x = 1(mod 3) 2x = 0(mod 6), or show that no such…

A: The given congruences are                                 x≡1 (mod 3)2x≡0 (mod 6)

Q: Prove that if a + x ≡ a +y (mod n), then x ≡y (mod n)

A: I have proved it by using defination of congruent.

Q: 1. Use Euler's Theorem to prove a = a (mod 105) for all a E Z.

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

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: Let n E Z. Prove that 3n # 1 (mod 9).

A: Proof by contradiction: Direct approaches can be difficult (or impossible) to use to prove a…

Q: Use mathematical induction on n to prove that if c≡d (mod m) then cn≡dn (mod m ).

A: Use the mathematical induction and the result If a ≡ b (mod m) & c ≡ d (mod n) then ac ≡ bd (mod…

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

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Q: Prove that for each integer a, if a not equal to 0 (mod 7), then a2 not equal to 0 (mod 7)

A: Consider the provided question,

Q: Show that for x and y integers, and m a positive integer x = y mod m implies x = y mod m.

A: This is a problem of Number Theory, Modular Arithmatic.

Q: Does x = a (mod p) always have a solution for every value of a, whenever p is prime?

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Q: Show that if a = b mod m and b = c mod m, then we have a = c mod m. Use the definition of congruence…

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Q: od k. If k and k2 are induced moduli for x prove

A: χ is a character mod k. k1 and k2 are induced moduli for χTo show: (k1,k2) is their gcd.

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: 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: 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 that whenever x ∈Zand d ∈Z+, if d does not divide (x2 −3x), then x mod d does not divide 3

A: The given statement is False.  Consider, x = 5 and d = 4. Then we see, x2 - 3x = 52 - 3x5…

Q: Exercise 25.2. Prove that log, (bc) = log, b+ log, c (mod p- 1) for all b, c not divisible by p.

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