... 9 2 7. Suppose p is an odd prime and a ‡ 0(mod p). For each k = 1, 2, . define ek and rk by where 0

Q: Prove that for all integers n ≥ 1, 10" = (-1)" (mod 11). Use part (a) to prove that a positive…

A: (a) Let n≥1 be an arbitrary integer. We know that 10≡10mod 11≡-1mod 11. Now consider 10n:…

Q: Find a number a such that 0 ≤ a < 111 and (10270+ 1)35 = a (mod 111).

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Q: s 19 a prime in the Eisenstein integers? is 79? If they are, explain why, if not, display a…

A: Given: Primes in Eisenstein integer. To find: a) To check wheater 19 and 79 are primes in Eisenstein…

Q: 2.4 Prove that if a and b are integers and p is a prime, then (a + b)² = ap+bP (mod p). You may…

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Q: rd. 17. Show that every decimal palindromic integer with an even number of digits is divisible by…

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Q: 7. Show that if n is odd, then p(4n) - 2 q(n). =

A: 7. Show that if n  is odd,then phi(4n)=2phi(n)

Q: Show that if n is an even perfect number, then n = 2P-'Mp for some Mersenne prime Mp 2P – 1.

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Q: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

Q: 1.16. Do the following modular computations. In each case, fill in the box with an integer between 0…

A: We have to calculate the following modulo.Note: According to the guidelines we are allowed to solve…

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: 13. a. Prove that for every integer n 1, 10" = (-1)" (mod 11). |

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Q: 13. Prove that if n ≥ 2, then 2" - 1 is not divisible by n. Hint: Let p be the smallest prime that…

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Q: 12.16. Give an example of a prime p of each of the forms: a. 4k + 1 b. 4k + 3

A: Let the prime number be

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

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Q: 1. Let p be an odd prime and g be a primitive root mod p. Show that Z = {gik : k = 1,...,p-1} if and…

A: please find the attached image.

Q: 5. Let n be a positive integer and p1,P2,P3, . „Pn be n distinct primes. If m is a positive integer…

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Q: 5. Prove that the integer 53103 + 10353 is divisible by 39, and that 111333 ible by 7.

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Q: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p &gt; 5. p2−1≡0mod24

Q: 1.21 Theorem. Let a natural number n be expressed in base 10 as  n=akak-1...a1a0 (Note that what we…

A: We know that 10 mod 3 = 1. So,

Q: mod 1537). 2. Frove the following statements: a) For any integer a, the units digit of a? is 0, 1,…

A: We can solve this expression of  numerical analysis

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: 38. It is known that 14 is a primitive root for the prime p = 30000001. Let 14⁹000000 (mod p). (The…

A: A primitive root mod p is a number a such that a has order p-1. This means p-1 is the least positive…

Q: (a pour) = (d(1) a) +...+ d& T dG T dl) +04+ mos ad ppo desta sedang

A: Here we use that (p-i)^p=-(i^p) in mod p. We proof this by binomial expansion.

Q: 13. Prove that if n ≥ 2, then 2 - 1 is not divisible by n.

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Q: 11. Prove metic them t of type

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Q: na ed t 4. Use the theory of congruences to verify that 25| (2"+4 + 33n+2 – 53n+6) for all n eN

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Q: 7 .Assune p is an odd prime such that p ≡ 1mod4 (that is p = 4k+1) Prove: if a is a primitive root…

A: Given: p≡1mod4       ∵p=4k+1

Q: 11.5. For each part, find an x that solves the given simultaneous congruences. (a) x = 3 (mod 7) and…

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Q: Suppose p is an odd prime. Show that (1² +2 +3P +...+ (p − 1)²) = 0 (mod p).

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Q: 3. Find the last two digits of the number 99°. [Hint: 9° = 9 (mod 10); hence, 99 = = 99+10k. notice…

A: In question number 3: we have to find the last two digits of the number (((9)^9)^9) That means to…

Q: Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no…

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Q: Let p be an odd prime and a, b are integers with pła, płb. Show that ab ()-(()) = Р

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Q: to seforo tebeg le v e b obo llo da e batainalo t sei o bo avd ba . 2. Let p be a prime number such…

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