3. Let d = (a, n) where n > 1. Prove that if there is a solution to ax = b (mod n), then d divides b.

Q: 6. Let m, n EN with m, n > 2, and let a, b E Z. Prove that a = b (mod mn) implies a = b (mod m) and…

A: (a) a = b (mod mn)       => there exists some integer k such that                    a = b +…

Q: I. Solve for the smallest value of x in the following linear congruences. 1. x = 3 mod 6 and x = 4…

A: To solve given linear congruences

Q: The linear congruence in two variables ax + by ≡ c (mod m), where a, b, c, and m are integers, m > 0…

A: As we know-the congruence ax+by≡c (mod m)has dm incongruent solution if   d=(a,b,c)|c& no…

Q: If mx = 1 (mod n) has a solution, then nx = 1 (mod m) has no a solution. Select one: True False

A: We know the definition of Congruence ,  Suppose m be a positive integer , we say that integers a and…

Q: 2. Give an example to show that a? = b2 (mod n) need not imply that a =b (mod n). 3. If

A: Given below the detailed solution

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

A:

Q: 4,75. Let n, m E Z. Prove that if n = 1 (mod 2) and m = 3 (mod 4), then n + m = 0 (mod 4).

A:

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: 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: (3) (2p = 2 (mod p), where (2) n! for 0 <k <n. k!(n - k)!

A:

Q: Suppose b is any integer., If b mod 12 = 7, what is 4b mod 12? In other words, if division of b by…

A: Given : b mod 12 = 7

Q: Show that: Ifp is odd prime of the form 4k+1, then find a solution of x +1 0 mod p

A: We shall use Euler’s Criterion here. Euler’s criterion states as follows: “Let p be a prime. The…

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

A:

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

A:

Q: Is the following proposition true or false? Justify your conclusion. For each a e Z, if a # 0 (mod…

A: The given proposition is as follows. For each a∈ℤ, if a≢0mod 3, then a2≡1mod 3. The given…

Q: Consider 64z 16 mod 196. The number of solutions mod 196 is Oa None Ob 3 Oe 1

A: We find the number of solutions incongruent mod 196

Q: Q3. Prove that if ax = ay (mod m) , then x = y (mod m/d) where d = gcd( m, a).

A: According to our policy at Bartleby, I can solve only one question. I am solving Question 3. Let's…

Q: Let n ∈ Z. Prove that if n^2 + 2n −4 doesnt equal 1 (mod 5), then 5 doesnt divide n.

A:

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

A:

Q: 22. Suppose c is an integer. If c mod 15 = 3, what is 10c mod 15? In other words, if division of c…

A:

Q: #5. Suppose that a = 2 (mod 113), and b = 5 (mod 113) What is (a b + 56b + 155) mod 113?

A: given a≡2(mod113),b=5(mod113)a3≡23(mod113)a3b≡235(mod113)56b≡280(mod113)56b≡54(mod113)155≡15mod113)

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)…

A:

Q: The last two digits of 52044 is 44. If ab = 0 (mod 3), then a = 0 (mod 3) or b = 0 (mod 3). F 10. F…

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: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: (18) If a, be Z, then (a + b)3 = a3 + b3 (mod 3).

A:

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: Ha = b (mod n) and gcd (a,n)= d, then gcd (b,n) %3D

A: Given a≡b(mod n). This can be written as a-b≡0(mod n). Implies a-b=kn for some natural number k.…

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

A:

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

A:

Q: (d) For all integers a and b, if ab = 7 (mod 12), then either a = 1 (mod 12) or a = 7 (mod 12).

A: (d) For all integers a and b, if ab=7 mod 12, then either a =1 mod 12 or a =7 mod 12

Q: If mx = 1 (mod n) has a solution, then nx= 1 (mod m) has a solution. Select one: O True False

A:

Q: Suppose b is any integer. If b mod 12 = 7, what is 6b mod 12? In other words, if division of b by 12…

A: To determine the value of 6b mod(12) if b mod(12) = 7.

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

A:

Q: A sequence of pseudorandom numbers is generated as follows Xo = 4 x₁ = ( 6x₁-1 + 5 ) mod 13 if i > 0…

A:

Q: Let n E Z. Prove that if 5n2 +3 =0 (mod 8), then 4 n.

A: In this question  we will be using contradiction concept so assume it is true statement then we…

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: 5. Let C = {x ∈ Z|x ≡ 7( mod 9)} and D = {x ∈ Z|x ≡ 1( mod 3)}. (a) If C ⊆ D? Justify your…

A:

Q: 10. Find the sum of all incongruent solutions of a =1 (mod 17). (Enter the least non-negative…

A:

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: e ab then th

A: Given: ak^2+bz^2 =c mod p To prove the given.

Q: For D = {1, 2, 3, 4, ..., 99}, find a counterexample to the claim, "For all æ, y € D, if æ mod 3 = y…

A:

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: Q3. Prove that if kx = ky (mod m) , then x = y (mod m/d) where d = gcd( m, k).

A:

Q: (b) Solve the set of congruences 2x =1 (mod 5) x=3 (mod 4)

A: Note:- We’ll answer the first question according our protocol. Please submit a new question…

Q: For which primes p > 7 is 7 a square mod p? Your answer should be of the form p = {a1, a2,... , a}…

A: Given: p&gt;7 ,If 7 is a square mod p.

Q: For any integers m and n, if m mod 11 is 6 and n mod 11 is 7, what is m+n mod 11? 13 9 42

A: