1.Let a, b, and c be integers, where a≠0. If a | b and a | c, then a | (b + c)            True or False

Q: The statement a ≡ b (mod n) is called a congruence.

A: Given that  The statement a ≡ b (mod n) is called a congruence.

Q: Q. Now make a multiplication table for modulo 3. ol 1 2 IN 0 1 2

A: Since you have uploaded multiple questions according to our guidelines we have solved first question…

Q: If x = y (mod 2) then which of the following is true. i. x and y are both odd ii. either x or y is…

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Q: perform the calculation in mod 7 a. [18]+[25] b. [5][8]

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Q: Let x be an integer coprime to 1450 that is not congruent to 1 modulo 1450. Suppose we know that x1…

A: The given two MCQs are exactly same. Hence we will just solve the first one. And we will use the…

Q: How do I prove this:  Let d ∈ N and let a, b ∈ Z. Prove or give a counterexample: If a 6≡ 0 (mod d)…

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Q: You are trying to factor n = 31831 which happens to be the product of 2 primes. Suppose your tions.…

A: An effective way to determine the greatest common divisor (GCD) of two integers is to use the…

Q: (a) 55 mod 7. (b) –101 div 3.

A: (a) 55 mod 7  As 55 = 7•7+6 So that  55 mod 7 = 6 mod 7

Q: 5. Let a, b, c, and n be integers. Prove that if a = b (mod n) and c = d (mod n), then a − c = b -d…

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Q: Which a. 24 b. 56 of the following is true? 3(mod4) 4(mod5) c. 62 = 4(mod4) d. 58 3(mod5)

A: We want to find true option.

Q: Show that for every integer n, either 3 | n or n² = 1 (mod 3).

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Q: Xavier and Zion are trying to calculate 7 + 3-1 . Here is their conversation: Xavier: "I plan to…

A: We know abc=a×c + bc a-b = a+(-b) We can switch the order of numbers in a addition problem but we…

Q: QUESTION 12 Show by contraposition that for all integers n, if „2 is odd, then n + 1 is even. For…

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Q: III. Fill-in the blanks. Tare more than 1 answer. Choose any. 1. 46 = 26 mod 2. 156 = mod 15 3. 41 =…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve the first three…

Q: Ans Encryt the message GO" by translating the letter, into number, applying the given encrytion…

A: We will use the basic knowledge of number theory and cryptography to answer this question completely…

Q: If a = 8(mod19) and b = 6(mod19), which of the following is the least residue of a × b modulo 19? ●…

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Q: If x98 = 1(mod 97) then possible value of x is of (a) 97 (b) 194 (c) 90 (d) 291 (е) None of these O…

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Q: Prove: The nth Fibonacci number Fn is even if and only if 3 | n.

A: The nth Fibonacci number Fn is even if and only if 3|n.

Q: Write the following expression in terms of In x, In (x + 1), and In (x + 2). 5 9 1 In X +2 x+ 1…

A: Solving the problem by just simplifying it.

Q: 1. A+B (mod 173) 2. C-D (mod 432) 3. 23 x E (mod F) 4. G+5 (mod 7) АВМ 1А A B C E F G Hannah 7550…

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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: 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: H 28. For all integers a, b, and c, if a b and a/c, then al(b + c).

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Q: 8. If a and b are odd integers, then 8|(a² – b²) -

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Q: 7. For all odd integers x and y, x² +3y² = 0 (mod8) . 8. The equation x* +x² +1= 0 has no real…

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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: 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: 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: - Show that (a + b)" = a + bn mod 2 for all integers a, b, and n with n > 0. Hint: Try looking at…

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Q: Ti If a = b (mod n) and b =c (mod n) then which of the following is true. (a) a = c (mod n) (b) a2 =…

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Q: Proposition 7. Let a, b, c, d E Z and n be a positive integer. If a = b (mod n) andc = d (mod n),…

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Q: Let a, n be positive integers. Knowing that 15a – 13n = 1, we deduce that the multiplicative inverse…

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Q: VI. Determine four values of x that will satisfy the following. 16. x = 4 mod 5 & x = 3 mod 7

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Q: true or false  1..The GCD of -50, -100 and -125 is -25 2.If (a, b) = d and d│m, then (ax is…

A: Solution

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: 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: C 1 a b (b²+ c²) mod 2 = ?, a, b, c EN - {0}

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Q: 5. Which of the following are inverses of each other? (a)3, 7 w.r.t. mod 20 (b)2, 5 w.r.t. mod 9…

A: Explanation of the answer is as follows