roblem 3. Recall that we showed that for any prime p > (;) -{ - p = ±1 (mod 8) p = ±3 (mod 8) 1 duce from this that (;) - 2 p2. 6) =(-1). on't forget to explain why is an integer. 8

Q: Prove this claim: For every positive integer d and every integer n, the following equation is…

A: Using division algorithm we solve this problem. The steps are given bellow.

Q: 3. Find integers x, y € [0,51] such that (a) 23x= 1(mod52) (b) 23y = 4(mod52)

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Q: Show that for all positive integers m and n, :(). m+n т+n m +n (m+1) . n. m т+1

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A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

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Q: Let x be an integer. Prove that, if x mod 9 = 5, then 9|(x^2-2x+3)

A: Given: x mod 9=5 then, 9|x2-2x+3|

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A: p is an odd prime.

Q: 2(5) – 3(1 – 6) 1. Is even integer? 5 2. List all even integers between -6 and 6. 3. Is 124/6…

A: Four questions are given. The solution to the four questions are given below.

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Q: For an integer n > 1, how many decimal stings of length n begin or end with an even digit? 5(10)"-1…

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Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

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Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

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Q: Please answer True or False for the following Statements... 1. For every odd integer n, the integer…

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Q: (Problem 6) Choose ALL correct statements, if more than one is correct. (A) [-5.76] = -5 (В) - 76…

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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: Every even integer greater than 2 is sum of two primes? Give your views on this?

A: We know that 2 is even prime. And all other primes are odd.

Q: (d) (e) (f) Assume that a 15 = 1 Find the value of ord41a. mod 41 and a #1 mod 41. How many…

A: As per the question we have to find : ord41(a) , where a15 ≡ 1 mod 41 but a ≠ 1 mod 41 total number…

Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Part III: Prove the following 5. Prove that 32.753215321... is a rational number.

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Q: 10. Prove that the following are irrational numbers: (1) √3 (ii) 1 + √2 (iii) √3+ √2 (iv)

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Q: Suppose p is an odd prime number, prove that p+1 (р — 3)! %3D (mod p) -

A: To prove: To prove that p-3!≡-p+12(mod p) for an odd prime number p.

Q: (g) 111... 111 (126 ones) (2) Prove that the numbers given in each part cannot be prime…

A:

Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

A:

Q: Suppose p and q are odd primes such that p = q + 8a for some positive integer a. () = () = ( ) .…

A: Legendre symbol:  For any prime number p and any integer a that is not divisible by p, the Legendre…

Q: 10.2 Find a number a that has the following three properties: (i) a = 73003 (mod 3750). (ii) 1 ≤ a ≤…

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Q: If x00 = 1(mod 109) then possible value of x is a) 108 (b) 218 (c) 109 (d) None of these

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Q: Q4 (a) Solve for x if 25x = 15(mod 29). (b) Determine the smallest positive integer so that when…

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Q: TRUE or FALSE: 1. There is no integer n such that 4 | (n^2-2) 2. The GCD of all prime numbers is 1.…

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Q: Find oll integers Such thot 24! x =1 Cmod 2) 5x = 3. (mod 8) and 1500 < X <2000

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Q: Prove: Let be odd prime. Prove that Aor all i with 1 LikP-2), you have an +... + working so far:…

A: As per the question we just have to show that the prime p divides the following : xi + x2i + ... +…

Q: 20.What is the value of n such that (24  is congruent to 58 mod n) ? a.13 b.17 c.-34 d.-13

A: # as per the guidelines we are entitled to solve one question at a time, please resubmit the other…

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Q: 1. i) Explain why 3 is a primitive root modulo 14 ii) What is log3(11) mod 14?

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A: This is a question of Number Theory.

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Q: Work the modular arithmetic problem. (12 9)(mod 3). (12 9)(mod 3) =(mod 3)

A: The given modular arithmetic problem is -  (12·9)(mod 3) We have to solve the above problem.

Q: 2 (-3) - = (- (-1)² 8 = {) 1 -1 if p= ±1 (mod 8) if p= ±3 (mod 8).

A: