Prove that ifa≡b(modn)andb≡c(modn), thena≡c(modn). Note that you must use thedivides definition of congruence in your proof, (that is,a≡b(modn)⇐⇒n|(a−b))

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

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

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Q: if a equiv b (mod n) , then a⁴ equiv b⁴ (mod n) for any positive integer 4.

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

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Q: If a= b (mod n) and gcd (a, n) = d, then gcd (b, n) = d :Select one True False

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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: TRUE or FALSE: 1. If a is congruent to b (mod m), then a^k is congruent to bk (mod m) for every…

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Q: What is the remainder when 2021202120212021 is divided by 2020? use congruence modulo

A: a = 2021202120212021 b = 2020 If a = qb + r then  a ≡ r mod n.

Q: (a) Show that if n 7 mod 8, then n + 6n 3 mod 8.

A: If, n = 7 (mod 8) So, n² = 7² (mod 8) So, n² = 49 (mod 8) So, n² = 1 (mod 8)

Q: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

Q: Compute the addition modulo operation 3+2 2 0. 1 3 O O

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Q: 3. Show that ifn = 2 (mod 4), then 9" + 8" is divisible by 5.

A: Here we will prove the given statement.

Q: [Hint: Proceed by induction on n.] 16. Use the theory of congruences to verify that 89| 24 – 1 and…

A: The solution is given as follows :

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: 2. In Section 3.1, we defined congruence modulon where n is a natural number. If a and b are…

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Q: a) Define what it means for a and b to be congruent modulo 7. b) Which pairs of the integers…

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Q: 4. (a) Solve the set of congruences х 1(mod 3), x= 2(mod 4), x = 3(mod 5).

A: We  will solve it by Chinese Remainder Theorem Given  x≡1(mod3)x≡2(mod4)x≡3(mod5) Compare it with…

Q: Consider the congruence 9x   21 (mod 33). (a) Does 9 -1(mod 33) exist? Explain why or why not.

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

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Q: What is 2 ^32 mod (5)?

A: mod is the function whose output is always positive. mod is represented by .

Q: Is 7|(15+20): True or False? 28 is congruent to 42 modulo 6: True or False?

A: I am solving these in step 2

Q: If a = 1 (mod n) and b = 1 (mod n) then a - b = 0 (mod n) اختر واحدا: 2 3

A: True.

Q: Which pairs of the integers -11, -8, -7, -1, 0, 3, and 17 are congruent modulo 7?

A: -11≡3 (mod 7) holds  Since -11-3 = -14 is divisible by 7 and leaves remainder 0. -7≡ 0 (mod 7)…

Q: (a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each…

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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: mod n, then x ≡ 0 mod n or

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Q: Prove directly from the definition of congruence modulo n that if a,c, and n are integers,n >1, and…

A: Let a, c and n be integers such that n>1. We have to prove that if a≡cmod n, then a3≡c3mod n…

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: proof that if a equiv b(mod n), then a+c equiv b+c (mod n) and ac equiv bc (mod n).

A: Apply the definition of congruence and divisibility, to express a≡bmod n. From the definition of…

Q: (b) Prove that multiplication mod n is commutative.

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Q: Prove directly from the definition of congruence modulo n that if a, c, and n are integers, n > 1,…

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Q: Use Chinese theorem to find all positive integers that makes this congruence true:        x = 2 mod…

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

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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: Use the definition of congruence to prove that if j, k, 1, m, nE Z are such that k =1 (mod n) and…

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Q: Q3/ Use Chinese theorem to solve the following congruence: X=2 mod 3 X=1 mod 5 X 2 mod 4

A: Since you have posted a multiple question ,I will solve the first question(i.e.Q3)  for you. To get…

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

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Q: Two integers are congruent modulo to a)prime integer b)odd integer C) not a prime nor composite…

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Q: Use a direct proof: p is a prime # and x ∈ Z. If x2≡x (mod p) then x ≡ 0 (mod p) or x ≡ 1 (mod p).…

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Q: Which are true for modulo 12? 1.) Does 5x ≡ 6 mod 12 have a solution? How many? 2.) Does 4x ≡ 6 mod…

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Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

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Q: If a (mod 0) = a, How would you interpret the congruence a ≡ b (mod 0)?

A: According to the given information, it is given that:

Q: If ab = 0 (mod m), then either a = 0 (mod m) or D=0 (mod m). Select one: O True False

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Q: If a # 0 (mod m) and b # 0 (mod m), then ab# 0 (mod m). Select one: O True False

A: justification is given in step 2)

Q: Let P be a prime, if ac ≡ bc (mod p) and p does not divide c, then a ≡ b (mod p) Select one: True…

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Q: (2) If a = b modn and a = b mod m and ged(m, n): 1. then a =b mod mn.

A: The greatest common divisor of two or more integers, which are not all zero, is the largest positive…

Q: Number theory

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