i +1 If z = E C, find simple expressions for i and /2 in terms of z. Then prove: If p is an odd prime, the numbers 2 and –1 are both squares mod p if and only if p – 1 is divisible by 8.

Q: 2. Euclidean Algorithm Use the Euclidean algorithm to find gcd(12345, 678).

A:

Q: Find Im(z) if z(8 + i) = 7+ (384 + 448i)

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

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Q: b=3(mod19)

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Q: The nth row of Pascal's triangle lists the coefficients of (x +y)^.

A: Here draw pascal triangle for some 1 to 4 rows and conclude for nth row

Q: Find all solutions in {0, 1,2, ..., 12} to x26 + 15x13 = 12 mod 13.

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Q: The modulus of 5 + 4i is * -41 -√41 √41 41 The conjugate of (i-i² )³ is -2-2i O 2+2i -2+2i Oi-1

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Q: N(a + bi)N(c + di) = N[(a + bi)(c+ di)] = 2. Show that 65 can be expressed as a sum of two squares…

A: See the detailed solution below.

Q: 2. Euclidean Algorithm Use the Euclidean algorithm to find gcd(12345, 678).

A: We need to find the GCD(12345,678)=?

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: (b) Compute 9-¹ (mod 19) by listing the multiples. Show your work.

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Q: (4 – 2æ)".

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Q: 1.34. This exercise begins the study of squares and square roots modulo p. (a) Let p be an odd prime…

A: Given:(a) Let be an odd prime number and let be an integer with does not divide . (b) (i) (ii)…

Q: Determine the cube root of (3 + j4). ound-off the modulus to 3 decimal places, the angle to the…

A: Cube root of (3+4j)

Q: x² + 4x = 10(mod11) (mod11) X=

A: Given that x2+4x≡10 (mod 11)x2+4x+4≡10+4 (mod 11)x2+4x+4≡14 (mod 11)x2+4x)(x+2)2≡14+11 (mod…

Q: Use mathematical induction to show that 32n – 1 is divisible by 8 for all positive integers n.

A: Solution

Q: Hellcopter Trip Whale Watching Adventure Volcano Tour Hours Cost ($) Hours Cost ($) Hours Cost ($)…

A: We can use linear equations to solve these types of problems. There are different methods available…

Q: Fill in the blanks in the proof of the of fact that 3" – 1 is a multiple of 2 for every natural n.…

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Q: (a) 64x 56 (mod 52) (b) 14x = 35 (mod 56)

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Q: Find a number a such that 70a = 1 mod 17.

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Q: Podasip 11. If p is prime, then every c can be expressed as a sum of two squares: x² + y² = c(mod…

A: The given is statement is not accurate. The correct statement in connection with the given statement…

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: 1-5w Зп — w Solve for w in terms of n in the formula

A:

Q: How do I use the Chinese Remainder Thm to solve this system of congrunces below? x ≡ 1 (mod 3) x ≡ 4…

A:

Q: True or False. Briefly explain why.  By Fermat’s Little Theorem, 6 is the smallest positive integer…

A: The given statement is FALSE. Fermat's Little Theorem states that if p is a prime number and a is an…

Q: Find the modulus and argument of 5 - 7i.

A: Since the modulus of a complex number a+ib is |a+ib|=a2+b2. So, the modulus of 5-7i is…

Q: Q4/ Find the element of GF(2³) then by using the multiplication table find (x+1)-¹ mod (x² + x + 1).…

A:

Q: 5x² + 3x = 3(mod17) X = (mod17)

A: Given that 5x2+3x≡3 (mod 17)x(5x+3)≡3 (mod 17)x≡3 (mod 17)  or   (5x+3)≡3 (mod 17)x≡3 (mod 17)  or…

Q: ind all solutions of the linear congruence: 9x = 5 mod (14)

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Q: 1. Div and Mod (a) Find the quotient ("div") and the remainder ("mod") when: (i) 23 is divided by 6…

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Q: 1. If a = 37 – 43 + k 2a - 36 a) 2 and 6 = -27+33 - k 7 b) a +56 express each of the following terms…

A:

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

A:

Q: Q7 Compute 59^11 mod 13 Note: You should be able to do this using the modulo squaring method and…

A: 59^11 mod13