Fermat's "Little" Theorem states that whenver n is prime and a is an integer, a"-l = 1 = 1 mod n a) If a = 463 and n = 953, then efficiently compute 463952 = mod 953

Q: The number 73 is prime. Enter an a such that 0 <a < 73 and a = 6 (mod 73), if such an a exists.…

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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: 4965 4 mod 5 4965 = 2 mod 7 4965 = 1 mod 11 Use your answers above to find the canonical…

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Q: 6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod…

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Q: For congruences of primes p = 1 (mod 4), a. If a is a quadratic residue modulo p, and p = 1 (mod 4),…

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Q: Suppose that 2n−1 ≡ 1 (mod n). Is n necessarily prime?

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Q: Show that if p is a prime number then every equation of the form ax ≡ b mod p with a 6≡ 0 mod p has…

A: Given p is a prime number and 6≡0 mod p. Therefore either p=2 or p=3. Let d=gcda,p. We know that if…

Q: Suppose a is an integer that is relatively prime to 400. Prove that a40 = 1 (mod 400).

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Q: Determine the smallest integer k > 1 such that ak = 1 modulo 2465 for all in- tegers a which are…

A: A composite number m is called Carmichael number if for all integers b coprime with m, bm-1≡1modm…

Q: Let p be an odd prime number and an integer ga primitive root with respect to mod p. So is the…

A: We will use basic knowledge of elementary number theory to solve this question.

Q: Use Fermat's theorem, show that if ged(a, 155) = 1 then a121 = a (mod 155).

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Q: 13. Compute 41023 (mod 11). a. 14. Prove that for any integer n, n31 = n(mod7).

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Q: For which odd primes p is 5 a square mod p?

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Q: a) Prove that 10" =+1 (mod 11) for any n EN. b) Suppose the integer r has digits rI-1I, I0. Prove…

A: Want to prove:(a) 10n≡±1 (mod 11) for any n∈N(b) If the integer x has digits xnxn-1.....x1x0 thenx≡0…

Q: Hello, I need help for the problem in the picture. Thank you!

A: We are given that α ∈ Z is a primitive root modulo an odd prime p. So,    αϕ(p) ≡ 1 mod p =&gt; αp-1…

Q: Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2q + 1 is also prime. Determine if 2 is a…

A: let p be a prime with p≡3mod 4 and suppose that q=2p+1 is also prime determine if 2 is a square mod…

Q: Let n be an integer greater than 1. If (n-2)! ≡ 1 mod n, then Prove n is a prime number.

A: Use the Wilson theorem and different case to solve this problem as in next step.

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: 6. Use Fermat's little theorem to compute : 3302 mod 7.

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Q: Find the least positive integer n that satisfies the congruence 7 ^128 congruent to n (mod13)

A: Given, 7128≡nmod 13 Find the value of n that satisfies the above congruence.

Q: 8. Use Euler's Theorem to find a number a between 0 and 9 such that a is congruent to 7 1000 modulo…

A: We know according to  Euler's Theorem  that  xϕ(n)≡1(mod n) where the function ϕ(n) counts the…

Q: Let P be a prime, if ac = bc (mod p) and p does not divide c, then a = b (mod p) Select one: O True…

A: Given  gcd(c, p) = 1 and ac ≡ bc(mod p). (We may assume p &gt; 1 so that c  ≠  0.) Then ac − bc =…

Q: Use Fermat's Little Theorem to find (Show work.) a) 945 mod.23. b) 231002 mod. 41 c) (-9)933 mod. 13

A: Fermat's Little theorem states that " if p is a prime number and a is not divisible by p then…

Q: 6. Show that ifn = pq, a product of distinct primes, then a (n)+1 (mod n) for all a. = a

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Q: 10. Use Euler's Theorem to show that 516) = 1 (mod 16).

A: Euler's theorem states that          "If gcd(a, n) = 1, then aφ(n) = 1 (mod n)

Q: 1. Suppose m, nEz* are relatively prime. Prove that for all a, bEZ, a =b (mod mn) if a =b (mod m)…

A: Note that: a is congruent to b modulo m if and only if m divides (a-b).

Q: Given any odd prime p, prove that: 2(p−3)! ≡−1 mod p. (Hint: Start with 2(p−3)!≡x mod p. Then work…

A: Let p be any odd prime number.  So we start with  2p-3!≡x mod p Also we have p-2≡-2 mod pp-1≡-1 mod…

Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: Prove that for any prime, p = 1 (mod 4) there are exactly (p-1)/4 incongruent values of a 0 (mod p)…

A: By theorem (a,p)=1 then xn≡a modp has g.c.d(n,p-1) solution or no solution according as…

Q: Fermat's "Little" Theorem states that whenver n is prime and a is an integer, d"-1 = 1 mod n a) If a…

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Q: Let m1, m2,…, mn be pairwise relatively prime integers greater than or equal to 2. Show that if a ≡…

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Q: The least residue of 5500 mod(17) is Select one: O none O 7 O 13 O10 O 8

A: The least residue is, 13 The detailed solution is as follows below :

Q: Fermat's "Little" Theorem states that whenver n is prime and a is an integer, d"-1 = 1 mod n a) If a…

A: Given, Fermat's Theorem :- If n is a prime number and a denotes an integer such that a,n=1, then…

Q: Consider the congruence mod 6 for integers. Z6={[0], [1], [2], [3], [4], [5]}. Prove the sum of an…

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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: show that If n is a positive integer and a is any integer prime to n, then a ln) = 1(mod n), where…

A: It's called Euler's generalization to Fermats theorem. I have used integer m instead of n.

Q: A.)Many printed airline tickets contain a 10-digit document number followed by a check digit. The…

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Q: 1. Let p and q be distinct prime numbers and let a be an integer not divisible by p and not…

A: As per BARTLEBY GUIDELINES , I answered one question (i.e. Q-1) and repost other questions…

Q: (c) Use Fermat little theorem to show that (a + b)® = a® + ® mod prime p and any positive integer a.…

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Q: Use Wilson's Theorem to prove that for primes p congruent to 1 modulo 4 the quadratic congruence 1'…

A: Given The quadratic congruence x2≡-1(mod p) To prove  x2≡-1(mod p) has a solution if p≡1(mod 4)…

Q: Use induction to prove for every positive integer, 4 x 10n = 4 or 7 (mod 33)

A: Suppose P(n) be any mathematical statement in terms of n. Then Principle of mathematical induction…

Q: Prove that if n is an odd positive integer then 1 + 2 + 3 + · · · + (n − 1) ≡ 0 mod n . What happens…

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Q: For which primes p > 7 is 7 a square mod p? Your answer should be of the form p = {a1, a2,... , a}…

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

Q: 6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not…

A: The given question is related with pseudoprimes. We have to show that if n = a2p - 1a2 - 1, where a…

Q: Suppose that Bob has used two distinct primes p and q to create the public modulus n = pq and public…

A: RSA algorithm: 1. Choose two distinct prime numbers p and q. Take n = pq. 2. Compute λ(n), where λ…

Q: Show that if p is a prime number then xy ≡ 0 mod p implies that x ≡ 0 mod p or y ≡ 0 mod p

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: