Let S be a set of non-negative integers with 80 elements. Show that we can find a subset of S such that the sum of its elements ( of this subset) is divisible by 80.

Q: Prove that any positive integer greater than or equal to 8 can be written as a sum of 3's and 5's.…

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Q: Prove or disprove that given a set of 125 integers, one can choose a subset of S of 16 of them so…

A: Given we have set which contains 125 different numbers. When we divide each number in set by 8 we…

Q: Let A be the set of five-digit numbers where only the digits 1, 2 and 3 are included. For example…

A: Given that A is the set of five-digit numbers where only the digits 1, 2 and 3 are included.  So,…

Q: Let x be the set of all positive integers and d(min) = | = - = | show that (x,d) is not complete

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Q: Suppose a, n E Z and 0 < a<n. Prove that )-(".")-(C ) n- 1 in = a a а — 1 in two different ways:…

A: We have to prove, na = n-1a+n-1a-1 First let us prove this using formula. Using Formula…

Q: Let $n$ be a positive integer. Among C(2n,0),C(2n,1),..., C(2n,2n), C(2n,n) is the largest. True…

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Q: 12.32. Give an example of a set S of four (distinct) positive integers such that the greatest common…

A: We have to find the required set S with the above property.

Q: e the binary operation on Z (set of integers) defined by a * b = 2ab +5 ow that + is commutative.…

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Q: Let a and b be non relatively prime positive integers. If there exist integers s and t with as+bt=5…

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Q: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

Q: of divisiblity.  For all integers a, b, and c, if a|b and a|c then a|(sb + tc) for any integers s…

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Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n&gt;1. All prime divisors of n!+1 must be bigger than n. This is…

Q: Let A be the set of words in the Oxford English Dictionary (OED). For each positive integer j,…

A: Assume a set A in which all of the words present in OED is present. Define a relation R such that, a…

Q: Let a, b be integers for which a01 and ba02 are relatively prime. Show that a and b are also…

A: Given: a and b are two integers for which a101 and b202 are relatively prime. We have to show that a…

Q: Find the smallest positive integer n, пп such that 0 and 0 are 2 coterminal. And give an example

A: Coterminal angles are the angles in the standard position having a common terminal sides.

Q: Show that if a|m and a|(ms+nt) for some integers a≠0, m, s, n, and t, then a|nt.

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Q: 4. Show that if n is a positive integer, then Σ {a₁,...,ak}{1,2,...,n} (Here the sum is over all…

A: Given, If n is a positive integer, then we have to show that ∑a1,.....ak⊆1,2,....,n 1a1a2....ak=n

Q: Prove that for any positive integers m and n there are m consecutive positive intege numbers such…

A: As per the question we have to prove that for any positive integers m, n there exist m consecutive…

Q: The set A consists of nine positive integers, none of which has a prime divisor larger than six.…

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Q: Discrete

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Q: prove that in any eight composite positive integers not exceeding 360, at least two are not…

A: Prove that in any eight composite positive numbers not exceeding 360, at least two are not…

Q: A binary operation * on a set of integers is defined as x*y = x^2 + y^2. Which one of the following…

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Q: b) Use generating functions to determine how many four element subsets of S = {1,2,3, ..,15}…

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Q: Let a and b be non relatively prime positive integers. If there exist integers s and t with as + bt…

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Q: Let X = {x;} be a set of n positive integers. Show that the sum of the integers of a subset of X is…

A: It is required to show that sum of elements of a subset of described set, X, is divisible by integer…

Q: Let A be the set of number in (3/5, 2/3) that have decimal expansions containing only finitely many…

A: Consider, Sn=0.66666.....666 ( where 6  appears n times). This is the largest n digit number in the…

Q: السؤال 2 2 The set P of positive integers has maximal element صواب ihi

A: Use th Archemedian property of set theory.

Q: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

A: According to the given question, if any positive integer n belongs to the required subset then it…

Q: Prove that there is an integer k for which 2021 = 2k + 1

A: To prove k is an integer  2021 = 2k +1

Q: Let A be a set of integers closed under subtraction. Prove that if A is nonempty, then 0 is in A.

A: Let A be a set of integers closed under subtraction. Prove that if A is nonempty, then 0 is in A.

Q: Prove that the set of all 2-element subsets of N is denumerable.

A: According to the given information, it is required to prove that set of all 2 elements subset of N…

Q: Let O denote the set of odd positive integers. Prove O is countable.

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Q: Prove that for every positive integer n, there are n consecutive composite integers.

A: This question is related to Descrete Mathematics

Q: Determine the number of ways a computer can randomly generate one or more such integers from 1…

A: There are 5 prime numbers between 1 to 12 . Which are 2 ,3 , 5, 7 , 11.

Q: Let Fin(N)(N) be the set of all finite subsets of NN. Prove that Fin(N)(N) is countably infinite.

A: A subset of N is finite if and only if it is contained in {0,1, ...,n} for some n∈N. Thus, we may…

Q: Prove that if n, k are positive integers with k | (n +4) and k| (n – 1), then either k = 1 or k = 5.…

A: k | (n+4)  k | (n-1)  =&gt; k | {(n+4) - (n-1)} =&gt; k | 5

Q: Find all positive integers n such that the sum of all positive divisors of n is odd.

A: Given the data

Q: 2. Show that there exists a proper subset A of N (i.e. AC N, but A ‡ N) such that A≈ N.

A: Given that A is a proper subset of ℕ To show: There exists a proper subset A of ℕ which is…

Q: If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which…

A: The number of r combinations which can be selected from n numbers if repetitions allowed is…

Q: Suppose that A is a set of 10 natural numbers, all of which are less than 50. Show that A must have…

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Q: Use mathematical induction to prove that for every integer n ≥ 2, if a set S has n elements, then…

A: To prove: For every integer n ≥ 2, if a set S has n elements, then the number of subsets of S with…

Q: 2) Let X be a finite set. Show by induction on n = = [X| that %3D |P(X)| = 2X1.

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Q: 7. Prove: If A is any set of 9 integers between 1 and 50, then there exist two different subsets XC…

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Q: Let a relatively prime positive integers. If there exist integers s and t with as + ht t - 11 then

A: if gcd (a,b) =d then there exists integers m and n such that d = a m + b  n  and d is the smallest…

Q: Let m, n be positive integers. If m n, prove that 5m - 1 5" - 1

A: We show that 5m-1|5n-1 whenever m|n ,for any positive integer m,n

Q: Let S be the set of positive integers less than 500. Find the number of integers in S that are…

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Q: If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which…

A: As we know that any truples h,i,j,k,m with n≥h≥i≥j≥k≥m≥1  can be represented as a string of n-1…

Q: Let n be any integer. Using a proof by cases, prove that n4 is either in the form of 5k or 5k+1 for…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.