true or false  1..The GCD of -50, -100 and -125 is -25 2.If (a, b) = d and d│m, then (ax is congruent to  b mod m) has exactly d incongruent solutions modulo m for any integers a, b and m with m> 0. 3.If g = (a, b) and a = gc, then (c, b) = 1 4.If (p, q) = 1 and ( q, r) = 1, then (p, r) = 1

Q: .3x = 4(mod5).

A: From the given expression of the modulus, we will have to find out the value of X. This is…

Q: Let c, a, B be nonzero real numbers such that c + B. Find a given that 92n 3 6. cn 4 ac? Türkçe: c,…

A: To find the value of a from the given summation.

Q: Q2: Find the solution of 143x³ + 22x¹ + 5x² + 3x + 1 = 0 (mod 11), if any.

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Q: et x, y E Z. We have (x+y)³ = (x+y)² (x+y) = (x² +2xy+y?)(x+y) = (x³+3x?y+3xy² + y³) = x³ + y3 od 3.

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Q: Let c, ax, be nonzero real numbers such that e. Find a given that ac² n=3 16 Ch Türkçe: c, a, 3…

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Q: 21 Q1/ If A = |3 -2 5|,then |A| is [2 [4 3 4 6. -120 О 10 О 110 * Q2/ If |A| = 3,then |A*| is O 12…

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Q: 1. Find all r €Z satisfying each of the following equations. (a) 3r = 2 (mod 7) (d) 9r = 3 (mod 5)…

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Q: Find the solutions of a3 – 24.x2 + 54 = 0( mod 25).

A: want to find solution of x3-24x2+54≡0(mod 25)note that1 a≡b(mod m) f and only if m|a-b(i.e a-b=mk…

Q: Informal Exercise 20. Observe that 4| 102. Show that to find Rem(n, 4), you just need to replace n e…

A: We need to find the remainder when n is divided by 4 .

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: a.) Determine if x< = 2 (mod 7817) has solutions. b.) Determine if x< = 3760 (mod 4861) has…

A: x2≡2(mod 7817) has a solution if and only if  2 is a quadratic residue of 7817. Euler's criterion…

Q: Let a and b be integers. Prove that GCD(a,b) divides GCD(a+b, a−b). Give an example when GCD(a,b)…

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Q: If 3x + 2 = 0 (mod 7), which of the following values can x assume? О а. -3 O b. 0 О с. 2 O d. -1

A: Given 3x + 2 = 0 (mod 7).

Q: Find the solutions of the congruence 15x2 + 19x ≡ 5 (mod 65)

A: The given equation is: 15x2+19x≡5(mod 65)

Q: For each of the FOllowing pairs OF numbers a and bi evaluate the ged (aib) and Find integers r and s…

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Q: Given a = (1, 2, 1) and b = (k, 1, 2) and a × b = √35. Determine the ax value(s) of k.

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Q: Let w = e2ni/23. Determine if 2 is prime in the domain Z+Zw+Zw²+• •.+Zw21. Justify your answer.

A: We will use group theory , especially the properties of cyclic groups to show that 2 is a prime in…

Q: Let c, a, ß be nonzero real numbers such that c + B. Find a given that 00 162n-3 ac² cn 4

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Q: Use calculus to sketch the graph of the given function. (a) f(x) = 3x* – 4xr

A: Consider the given function, fx=3x4-4x3 Here we need to sketch the graph of the given function using…

Q: For each pair of integers m and d, calculate m mod d. Show your work! a. m = 43 and d = 9 b. m 50…

A: If m and d be two integers . Then m mod d means that the remainder when m is divided by d .

Q: Find the solutions of the congruence x2 ≡ 16 (mod 105).

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Q: y²=x³-5x+3 (mod5). solve for all x and y, when x is at least 0, and y is at most 4

A: Using congruence and modular arithmetic we solve the given equation. We substitute the value of x…

Q: Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can…

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Q: 3. Let a be a primitive 10th root of unity. Which of the following is false? A. a5 = 1 B. a³ = a¹3…

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Q: Q7. Solve the following set of simultaneous congruences. 6x = 1(mod 5) 5x = 3(mod 11) 2x = 5(mod 13)

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Q: State whether True or False: If A and B are integers and A ≡ 3 (mod 3) and B ≡ 2 (mod 6) then A+B ≡…

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Q: -1 21 2 If A = [], B = [3], and C = [12], find 0 0 2. 4 A² + BC.

A: Step 1: Step 2: Step 3: Step 4:

Q: Give all non-negative solutions for 10r =-17 mod 4.

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Q: It is a fact that if n is any nonnegative integer, then 1-(1/2"+1) 1 1 1 1+=+ + + •+ 2 2² 23 1-(1/2)…

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Q: 1. Let a and b be integers. Prove that GCD(a,b) divides GCD(a+b, a−b). Give an example when GCD(a,b)…

A: Let, GCD(a, b) = d1 =&gt; d1 | a and d1 | b=&gt; d1 | (a + b) and d1 | (a - b) =&gt; d1 is a common…

Q: or what values of k, x=4 will be a solution for 3x²+4x = k mod31?

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Q: Prove that it is impossible to choose three distinct integers (a – b)|(b – c), (b – c)|(c – a), (c –…

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Q: Exercise 2.6. Prove the following. For each, m and n are integers. Exercise 2.5. Prove the…

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