Informal Exercise 20. Observe that 4| 102. Show that to find Rem(n, 4), you just need to replace n e N with the number formed from the last two digits of n. Show that if di and do are the last two digits, then n = 2d1 + do mod 4.

Q: 2p a) n! = 2 (mod p), where for 0 <k < n. %3D k!(n - k)!

A: To prove the given part (a) 2pp≡2mod p where nk=n!k!n-k! for 0≤k≤n

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A: a ≡ b (mod 17) means that a − b = 17k for some k ∈ ℤ. ∴  b − a = −17k = 17(−k); henceb ≡ a (mod…

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Q: A1.b. Find the inverse of17 modulo 23, if exists. If not, give reasons.

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Q: True or False : a – c ≡ b – c (mod n) ⟺ a ≡ b (mod n)

A: We have to solve given problem:

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Q: [Hint: Proceed by induction on n.] 16. Use the theory of congruences to verify that 89| 24 – 1 and…

A: The solution is given as follows :

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A: The given proposition is as follows. For each a∈ℤ, if a≢0mod 3, then a2≡1mod 3. The given…

Q: Al. b. Find the inverse of 17 modulo 23, if exists. If not, give reasons.

A: The objective is find the inverse of 17 modulo 23 if exists or else give the reason why it does not…

Q: (d) Using the result in (c) find the inverse of 45 mod 17.

A: (d) we see that, 17×8 + 45×(-3) = 1, Where, p = 8 and q = -3

Q: 2. Give an example to show that a (mod n). b (mod n) need not imply that a =b

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Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

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Q: côngri 8.8. Use the technique of the previous problem to solve the following systems of congru-…

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Q: Suppose you are using the fast method to compute xe mod N, where x=23, e=101 and N=515. You have…

A: We will find out the required value.

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Q: show that 2 is a primitive root mod 3^e for all e>=1

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Q: Recall the Fibonacci numbers Fn are defined by Fo = 0 , F1 = 1 and Fn Fn-1 + Fn-2 for all n > 2 .…

A: Consider the provided question, Fibonacci numbers Fn are defined by F0=0, F1=1 and Fn= Fn-1+Fn-2 for…

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Q: 3. The Lucas Lehmer Test is applied below to test whether the number 255 is prime or not. (S,' -2 k…

A: The Lucas Lehmer test: Let Mp =2p-1  be the Mersenne number to test where p is an odd prime.…

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Q: Σ- n+ 1) (2n + 1), n2 1 i=1

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Q: 5 how that Cm d n+2 Che prine 4C(n-1)! +)+n = 0 mod @ncnt2)

A: We show that the n and n+2 are primes If and only if  4((n-1)!+1)+n≡0 mod n(n+2) We take n as prime…

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A: Consider the given equation.

Q: (d) Justifying your answer, find all integers r satisfying the two simultaneous congru- ences x = 1…

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Q: 3. If a = b (mod n), prove that gcd(a, n) = gcd(b, n). 4. (a) Find the remainders when 250 and 4165…

A: Given below the detailed solution

Q: b) 100 hours aft 24. Which decimal digits occur as the final digit of a fourth power of an integer?…

A: We have to find decimal digits occur as the final digit of a fourth power of an integer.

Q: Consider the statement P given by: "For all n E Z, if n mod 4 1, then n2 mod 4 = 1." (a) Is P true…

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Q: Let x, y, N E N be n-bit integers such that N > 2. . If x < N, the value x" mod N can be computed…

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Q: 5. By the principle of nathematical inductionprove that 3antl +(-D"2=O(mod 5) forall positive…

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Q: 2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35). [Hint: From Fermat's theorem a = 1 (mod 7) and…

A: Fermat's Little Theorem:- It states that if p is any prime and a is any integer then, ap≡amodp…

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A: We have to prove that Given statement

Q: For n=195, by Fermat's theorem a-2 a) a' (modn) b) 1(modn) c) 0(modn) d) a(mod n)

A: Fermat's theorem

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A: Given: p&gt;7 ,If 7 is a square mod p.

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Q: D. Define the multiplication operation * as modulo 10 in G = {1, 3, 5, 7,}. Cr table for this.…

A: Topic:- algebra

Q: 5. Show that (a + b)n = a + bn mod 2 for all integers a, b, and n with n > 0. Hint: Try looking at…

A: Given data: (a-b)n= an +bn mod 2 for all intergers a,b and n with n&gt;0.