1. Show that if pla + a + a? + a+1 and a -1 mod 5 then P=1 mod 5. Hint: a -1= (a - 1)(a* +a + a? + a+1). What is the period of powers of a when seen modulo p? 2. Show that there are infinite many primes p such that pl mod 5. Hint: if p, P , Pa are the first n primes consider A + A+ A? + A'+1 where 10p, PPs.. Pa-1. A =

Q: Prove that there are infinitely many primes congruent to 3 mod 8. previous part.

A: We have to show that there are infinitely many primes congruent to 3 mod 8. we will proof this by…

Q: 3400 m is prime numbers, then compute 27 (mod m) 2. If m is natural numbers, show that a (mod m ) =…

A: Modular arithmetic: a≡b mod c means c divides a-b i.e., c|(a-b). So there exists a constant k such…

Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

Q: Please give detailed steps!

A: :

Q: Find an integer n such that n = 1 mod 7, n = 3 mod 8, and n = 2 mod 9.

A: We can solve this system of congruences using the Chinese Remainder Theorem.

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: (()) b) Use part a) to find an integer that satisfies the congruence x² = -1 (mod 17) a) Prove that…

A:

Q: Show that if p is an odd prime and a and b are integers with a ≡ b (mod p), then

A: According to the given information, let p be an odd prime and a and b are integers with a = b…

Q: The remainder when a is divided by 7 is 2 and the remainder when b is divided by 7 is 5. (So a mod 7…

A: Given that remainder when "a" is divided by 7 is 2

Q: 2. Find a primitive root modulo 41.

A: We have to find a primitive root of 41.

Q: Consider the sequence an = in + (-i). Even though the formula for a refers to complex numbers, all…

A:

Q: 3 Р 1 ? What if 3 is replaced by = 2. For which primes p is -3, 5, -5, 6, or -6 ? Any patterns? Is…

A: The symbol  is the Legendre symbol, used in number theory to denote whether a number is a quadratic…

Q: 2221 (mod 13) is equal to O 10 (mod 13) O 1 (mod 13) 7 (mod 13) 6 (mod 13) O None of these

A:

Q: 5. Using congruence theory (not brute force), find all solutions to the following linear congruence:…

A: Note:- A two variable linear congruence ax+by≡m    mod (n) has solution if and only if  gcd ( a, b,…

Q: 7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0…

A:

Q: a) If a| (21b−c) and 6c≅3b(mod2a), then a| 41c b) If a≅b(modc) and d|c then ?≅?(mod?) c) ∴(d|…

A: this is a question of modular arithmatic.

Q: Suppose that p is a prime number. Given this information prove that if x is an integer such that x^2…

A:

Q: Najabhai

A: The objective of the question is to find all integers n that satisfy the equation n² + 503 = 0 (mod…

Q: For n a nonnegative integer, either n = 0 mod 3 or n = 1 mod 3 or n = 2 mod 3. In each case, fill…

A:

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

A:

Q: Determine the possibilities for the final digit of a sixth power of an integer.

A: The last digits of a number are 0,1,2,3,4,5,6,7,8 and 9. The number 0 raised to any number will be 0…

Q: 4. Let p> 2 be prime. Show that there exists an integer n such that n² + 1 is a multiple of p if and…

A: We use the fact that if gcd(a,p)=1 then a^{p-1}= 1 (mod p)

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.