
For any two positive integers a, b, define k(a,b) to be the largest k such that a* | b but ak+1b. Given two positive integers x, y, show that (a) k(a, gcd(x, y)) = min{k(a, x), k(a, y)} for any positive integer a (b) k(a, lcm(z, y)) = max{k(a,a),k(a, y)} for any positive integer a. Hint: Think of the prime factorization of the numbers
For any two positive integers a, b, define k(a,b) to be the largest k such that...
Recall we say that a Pythagorean triple is a triplet of positive integers a, b and c such that a^2 + b^2 = c^2. Examples are (3,4,5) and (5,12,13). Explain why for any Pythagorean triplet one must have that 12 divides abc. (Hint: It may be easiest to do this by showing that 3 divides abc and showing separately that 4 divides abc).
Recall that for any integers x and y, we say that x is divisible by y if and only if there exists an integer k such that x=ky. Prove by induction the following claim: n^3 + 2n is divisible by 3. ( n^3 =n*n*n)
Given a set S of integers, we say that S can be partitioned if
it can be split into two sets U and V so that considering all u U
and all v V, u = v. Let PARTITION = { <S> | S can be
partitioned }.
a. (5) Show that PARTITION NP by writing either a verifier
or an NDTM.
b. (15) Show that PARTITION is NP-complete by reduction from
SUBSET-SUM.
9.(20) Given a set...
Suppose that we have two unsorted lists of integers A and B. The lists are the same size, n. a) Write an algorithm that outputs how many integers occur in both lists. An integer occurs at most once in each given list. For example: if A = [1,2,5,7,13,19] and B = [2,9,13,14,19,22], then we can see that elements {2, 13, 19} occur in both lists, so the output will be 3. b) If the lists were sorted, say how we...
9. (20) Given a set of integers, we say that can be partitioned if it can be split into two sets U and V so that considering all u EU and all v € V, Eu = Ev. Let PARTITION = {<s> S can be partitioned }. a. (5) Show that PARTITION € NP by writing either a verifier or an NDTM. b. (15) Show that PARTITION is NP-complete by reduction from SUBSET-SUM.
Given a set S of integers, we say that S can be partitioned if it can be split into two sets U and V so that considering all u U and all v V, u = v. Let PARTITION = { | S can be partitioned }. a. (5) Show that PARTITION NP by writing either a verifier or an NDTM. b. (15) Show that PARTITION is NP-complete by reduction from SUBSET-SUM.
Given a set S of integers, we say that S can be partitioned if it can be split into two sets U and V so that considering all u ∈ U and all v ∈ V, Σu = Σ v. Let PARTITION = { <S>| S can be partitioned }. a. (5) Show that PARTITION ∈ NP by writing either a verifier or an NDTM b. (15) Show that PARTITION is NP-complete by reduction from SUBSET-SUM.
Given a set S of integers, we say that S can be partitioned if it can be split into two sets U and V so that considering all u Î U and all v Î V, Su = Sv. Let PARTITION = { | S can be partitioned }. Show that PARTITION is NP-complete by reduction from SUBSET-SUM
QUESTION 19 Let P(m, n) be the statement "m divides n", where the domain for both variables consists of all positive integers. (By “m divides n” we mean that n = km for some integer k.). is an Vm P(m,n). O a. False b. "False" and "not a tautology" O c. True d. Not a tautology QUESTION 23 Let P(m, n) be the statement "m divides n", where the domain for both variables consists of all positive integers. (By “m...