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• ### 1. Let n be a positive integer. Classify the languages (i) R = {(M)IM is a...

1. Let n be a positive integer. Classify the languages (i) R = {(M)IM is a TM and L(M) contains exactly n strings) (ii) S- (M)|M is a TM and L(M) contains more than n strings as (a) decidable, (b) Turing-recognizable but not co-Turing-recognizable, (c) co-Turing-recognizable but not Turing-recognizable, (d) neither Turing-recognizable nor co-Turing-recognizable. Justify your answers.

• ### Let n be a positive integer. For each possible pair i, j of integers with 1...

Let n be a positive integer. For each possible pair i, j of integers with 1 sisi<n, find an n x n matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.

• ### URGENT Question 3 25 pts ArrayMystery: Input: n: a positive integer Pseudocode: Let output be an...

URGENT Question 3 25 pts ArrayMystery: Input: n: a positive integer Pseudocode: Let output be an empty array For i = 1 to n j = 1 While ij <= n Addj to the end of output j - j + 1 Return output Answer the following questions about the ArrayMystery algorithm above. a) How many times will the inner while loop iterate? You should express your answer in terms of i and n, using Big-Oh notation. Briefly justify your...

• ### Let n be a positive integer. For each possible pair i, j of integers with 1<i<i...

Let n be a positive integer. For each possible pair i, j of integers with 1<i<i <n, find an n xn matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.

• ### 1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers...

1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...

• ### Let no be any integer (positive, negative, or zero). Let Pno) Pno+1, tions, one for each...

Let no be any integer (positive, negative, or zero). Let Pno) Pno+1, tions, one for each integer n > no, such that .., Pn,.. - be proposi- |(a) Pn is true (b) for each integer n > no, Pn implies P n+1 Let no be any integer (positive, negative, or zero). Let Pno) Pno+1, tions, one for each integer n > no, such that .., Pn,.. - be proposi- |(a) Pn is true (b) for each integer n > no,...

• ### Q18 12 Points For any positive integer n, let bn denote the number of n-digit positive...

Q18 12 Points For any positive integer n, let bn denote the number of n-digit positive integers whose digits are all 1 or 2, and have no two consecutive digits of 1. For example, for n - 3, 121 is one such integer, but 211 is not, since it has two consecutive 1 's at the end. Find a recursive formula for the sequence {bn}. You have to fully prove your answer.

• ### 1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm...

1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm we can write m=qn+r, with 0 <r<n-1. As in class define (m,n) = {mc+ny: I,Y E Z} and S(..r) = {nu+ru: UV E Z}. Prove that (m,n) = S(n,r). (Remark: If we add to the definition of ged that gedan, 0) = god(0, n) = n, then this proves that ged(m, n) = ged(n,r). This result leads to a fast algorithm for computing ged(m,...

• ### (a) Let n be any positive integer. Briefly explain (no formal proofs) why n > 1...

(a) Let n be any positive integer. Briefly explain (no formal proofs) why n > 1 ≡ ¬(n = 1). (b) Recall that a positive integer p is prime iff there do not exist a positive integers n and m, both greater than 1, such that p = nm. (I.e., Prime(p) means ¬∃n ∃m (n > 1 ∧ m > 1 ∧ p = nm).) Give a formal proof of the following: for any prime p, any positive integers n...

• ### I got a C++ problem. Let n be a positive integer and let S(n) denote the...

I got a C++ problem. Let n be a positive integer and let S(n) denote the number of divisors of n. For example, S(1)- 1, S(4)-3, S(6)-4 A positive integer p is called antiprime if S(n)くS(p) for all positive n 〈P. In other words, an antiprime is a number that has a larger number of divisors than any number smaller than itself. Given a positive integer b, your program should output the largest antiprime that is less than or equal...