Let m be a positive integer not divisible by 2 or 5. Prove that the decimal expansion of the fraction 1/m is periodic with period equal to the order of 10 modulo m.

Q: Consider the linear congruential number generator given by n + (ахр-1 + с) mod m

A: In this question, we have given a linear congruential  generator xn←axn-1+c  mod(m)Given thata=4,…

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

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Q: bute the order of 10 modulo 13. Compute the period of th /13.

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

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Q: Let p be an odd prime and let g be a primitive root modulo p. Prove that a has a square root modulo…

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Q: 6. Prove that m is prime if and only if y(m) = m - 1.

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Q: 5. Show that 2 is a primitive root mod 61 and compute log₂ (29) mod 61.

A: To show 2 is a primitive root we have to show that it has order 60. We do this and illustrate the…

Q: SImalest posiuiy OI the n) is order of a modulo n. Find the additive orders of each of the following…

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Q: 1. Compute the least nonnegative residue of 10k + 1 modulo 13 for k 1, 2, 3, 4.

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Q: Let A = 1 = {n+ (−1)n 2" | REN}. n Compute sup A and inf A if they exist.

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Q: Let n be an element in Z. Prove that 5|n^2 if and only if 5|n.

A: If n be an element in Z. We have to prove that 5|n^2 if and only if 5|n.

Q: Solve for d, 19d ≡ 1 mod (37422000) using the "Extended Euclid's Algorithm"

A: The solution is given by using Extended Euclid's Algorithm as follows

Q: 6. Let Fn 22" + 1, n > 1, be a prime Fermat number. Show that 3 is a primitive root modulo Fn.

A: Given: Fn=22n+1,n>1

Q: Prove that xk = x* mod |G| for x € G for any integer k.

A: Prove it considering two cases.

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: Prove that Fisher's ideal index number lies between Laspeyre's and Paasche's index number

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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: Ulia primeive Toute 95. Find all primes p ≤ 29 such that 2 is a primitive root modulo p.

A: Here we have to find all primes less than 29 or equal to 29 having primitive root 2.