Fermat's Little Theorem tells us that if p is prime and a is an integer not divisible by p, then aP-l=1(mod p) . Use this result to find the remainder when 13302 is divided by 31. Please note that both 13 and 31 are prime numbers.

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

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Q: Give an odd prime p congruent to ±1 modulo 12. Show that (3) = 1.

A: Please see the attachment

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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…

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Q: Let be a prime number. Stating any results that you use, deduce that xº-1 = (x – 1)(x – 2) · . · (x…

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Q: Where would 129 be on a mod 13 clock (which goes from 0 to 12)? What is the remainder…

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Q: Two great mathematicians who have contributed greatly in the field of modular arithmetic are Swiss…

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Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

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Q: Using Fermat's Little Theorem find 52003 mod 455. Note that 455 is not a prime number and 455…

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

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Q: Find the remainder using Fermat’s Little theorem 452 mod 11.

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Q: Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1.…

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Q: 3. Use Wilson's Theorem to prove 6(k - 4)! = 1 (mod k:), if k is prime.

A: Wilson's Theorem: Let k be a prime number, then by Wilson's theorem, (k-1)!=-1(modk)

Q: Suppose b is any integer. If  b mod 12 = 5,  what is 6b mod 12? In other words, if division of b by…

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Q: What prime numbers are squares in mod 17? (Include primes that are larger than 17). What primes p…

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Q: Suppose n is a large odd number. You calculate 2(n-1)/2 = k (mod),  where k is some integer k is not…

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Q: If you randomly pick integers from 1 to 200, how many would you have to pick to ensure that at least…

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Q: Using Euclid's Algorithm, what is a multiplicative inverse of 23 mod 75 and then solve 23x is…

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Q: There exist two prime numbers and such that their product gives 1261

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

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Q: a) Show that 2340 ≡ 1 (mod 11) by Fermat’s little theorem and noting that 2340 = (210)34.b) Showthat…

A: According to the given information, Fermat’s little theorem says that:

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: Characterize the primes p mod 20 such ial

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Q: 5) Using Fermat's little theorem, determine 21998 mod 7. In other words, what is the remainder when…

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Q: Prove by Induction that for every natural number n, 3n-1 is a multiple of 2? Attach File

A: Introduction: Divisibility is a very important property in the set of integers. A number can either…

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: Suppose c is an integer. If c mod 15 = 3, what is 10c mod 15? In other words, if the division of c…

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Q: Euler's Theorem states that if n is a positive integer and a is relatively prime to n, then ar(n) =…

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Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

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Q: 122 is congruent to 5 modulo17 ?

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

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Q: True or False For arbitrary integers a and b, a ≡ b(mod n) if and only if a and b do not have the…

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Q: Show that the product of two numbers of the form 4n + 1 is still of that form. Hence show that there…

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Q: 3. Use Euler's Theorem (and the Chinese Remainder Theorem) to how that n12 = 1 (mod 72) for all (n,…

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Q: Using Fermat's theorem, find 3201 mod 11.

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Q: Use Fermat's theorem to Reduce 50250 (mod 83) to a number in the range {0,1

A: Here we will find out the required value.

Q: Prove that the number in between twin primes is always a multiple of 6.

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Q: 2. Find an inverse of 2 modulo 17 that is positive and less than 17.

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Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

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Q: Find a number that is less than 1740719 which is congruent to 577 modulo 1279 and congruent to 317…

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Q: a) Use Fermat’s little theorem to compute 3302 mod 5, 3302 mod 7, and 3302 mod 11.b) Use your…

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Q: Construct the addition and multiplication table for Z4 (Note that, both operations are mod4)

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Q: 8. Show that 18!= -1 (mod 437). 9. Using Euler's Theorem, find the last two digits of 44443333 10.…

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Q: Suppose b is any integer. If b mod 12 = 7, what is 6b mod 12? In other words, if division of b by 12…

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Q: Find the least residue of 99! (Mod 101)

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Q: Fermat's theorem is not true for some prime number. اختر واحد

A: False

Q: How would you find -88 mod 13?

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