24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

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

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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: 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: (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: Prove that if ax = b (modn) then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x =…

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Q: 1. How many solutions( mod 419) does x = 186 mod 419 have? (Show your work!) 2. How many solutions (…

A: Given: x2≡186 mod 101 We need to solve x2≡186 mod 101 and how many solution exists.

Q: 5- If a = b mod(n), then ac = bc mod(n). 6- If a b mod (n), then ak = b*mod(n) for every positive…

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Q: He have to Show that lon = (-1)" (mod 1") shall use mathematical moluction Let Pin) := lon a (47"…

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Q: 13. (a) Prove that for each integer a, if a 0 (mod 7), then a #0 (mod 7).

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Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

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Q: The set {0,5,6} is closed under Mod(10)

A: To check if { 0 5, 6} is closed under mod(10)

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: 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: 2

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

A: Given  gcd(c, p) = 1 and ac ≡ bc(mod p). (We may assume p > 1 so that c  ≠  0.) Then ac − bc =…

Q: 1. Show that an integer n is divisible by 2 if and only if its last digit is even. (Hint: For each i…

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Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

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Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

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Q: Find the smallest positive integer for which x mod 3=2 and x mod 4=3   What is the next smallest…

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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: Show that y = x (mod p) leads to -y = -x (mod p).

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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: Consider the congruence mod 6 for integers. Z6={[0], [1], [2], [3], [4], [5]}. Prove the sum of an…

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Q: (4) Let p be a prime number. Then r = 1 mod p has exactly two solutions modulo p.

A: 4) To prove, x2≡1mod p has exactly 2 roots such that p is a prime number.

Q: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

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Q: Show that if a is an even integer then a² = 0 (mod 4) and if a is odd integer then a² = 1 (mod 4)

A: We know that an even number is a multiple of 2 and we write an even number k as k=2m for some…

Q: . How many different values are there for 4" (mod 7), as n ranges over all positive integers?

A: Note - As per our guidelines we have to answer first question if multiple questions are posted. So…

Q: 3) If the gcd(z,n) = d and xz = yz mod n then: Show that x = y mod

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Q: 10. Use Euler's Theorem to show that 5P(16), '=1 (mod 16). Show your work or explain your reasoning…

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

A: First divide 1 i.e. the dividend by 6 i.e. the divisor 1/6 = 0.16667 Then multiple the whole part of…

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: Let x, y ∈Z. Prove that if x≡1(mod 5) and y ≡2(mod5 ), then x^2-y^2≡0(mod 5).  Give 2 examples to…

A: Given, x,y∈ℤ and x≡1mod 5, y≡2mod 5

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: 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: Show that if x mod 8 = 1, then (3x −5) mod 6 = 4.

A: Solution

Q: Prove that if n is an odd positive integer then 1 + 2 + 3 + · · · + (n − 1) ≡ 0 mod n . What happens…

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Q: Prove or disprove: for any integers a, b, c, d, if a= c (mod n) and b= d (mod n), then ab = cd (mod…

A: Given: For any integers a, b, c, d if a≡cmod n and b≡dmod n To show: ab≡cd (mod n) To say that a is…

Q: Prove the above property. That is, prove that if a mod n = k and b mod n = j, then (a + b) mod n =…

A: If a mod n =k and b mod n=j show that a+b mod n =j+k mod n. Definition: If x mod y=z, then y | x-z.…

Q: What is the smallest positive integer x in the equarion: 1=3x+2(mod 7)?

A: We have to solve the equation 1 = 3x + 2 (mod 7)⇒1 - 2 - 3x = 0 (mod 7)⇒-1 - 3x = 0 (mod 7)⇒3x = - 1…

Q: Show that if p is a prime number then xy ≡ 0 mod p implies that x ≡ 0 mod p or y ≡ 0 mod p

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