any set of 201 integers must contain at least 21 elements that all have the same last digit.

Q: Write a complete residue system modulo 13 consisting only of odd integers.

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Q: Let S be a set of non-negative integers with 80 elements. Show that we can find a subset of S such…

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Q: How many positive integers between 500 and 999 inclusive are not divisible by 5?

A: For this problem, we must recall that to calculate the number of integers in a range of values that…

Q: Prove that for all natural number n, 52n-1 is divisible by 8.

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Q: Establish by mathematical induction that any postage of at least 12c can be obtained using 3c and 7c…

A: We have to establish by mathematical induction that any postage of at least 12c can be obtainedusing…

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: Any integer greater than 1 that has other positive factors besides 1 and itself is also known as O…

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Q: Any two integers are congruent modulo to: not a prime nor composite integer o even integer odd…

A: Let a, b are two integers. Two integers a, b are congruent modulo to n (where n is an integer) if…

Q: Show

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Q: Let N be a set of all positive integers with relation of divisibility. List all immediately…

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Q: Determine the number of permutations of the set {1 ,2 ,..., 14} in which exactly 7 integers are in…

A: Solution-:

Q: Prove: If x is a positive prime integer, then x+7 is a composite integer. (i.e., not prime)

A: See solution below

Q: Prove there are infinitely many prime numbers

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Q: Find how many positive integers with exactly four decimal digits, that is, positive integers between…

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Q: Give three elements of the equivalence class of 15 modulo 7

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Q: How many triples of integers from 2 through 16 can be formed in which the integers are in increasing…

A: Any triple of integers (i, j, k) with 2 ≤ i ≤ j ≤ k ≤ 16 can berepresented as a string of 15 – 1…

Q: Compute the sum of all integers between 60 and 150 that are exactly divisible by 8.

A: Consider the given range of number 60 to 150

Q: Let Sbe the set of all (nonnegative) 4-digit integer. How many integers from start with 1 and end…

A: let S be the set of all 4-digit integer. how many integers from start 1 and end with 9

Q: Prove that there are infinitely many prime numbers

A: Let us assume that there exist finitely many prime numbers (Let n). Here, n is a positive integer.…

Q: Discrete

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Q: How many positive integers less than 10000 are there which contain at least one 2 or at least one 3…

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Q: Compute the sum of all integers between 40 and 100 that are exactly divisible by 3.

A: Compute the sum of all integers between 40 and 100 that are exactly divisible by 3.

Q: Prove a number is divisible by 3 if and only if the sum of its digits is divisible by 3.

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Q: Show that, selecting 91 positive integers guarantees that there are at least 10 integers with the…

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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: 5. Give a recursive definition of the set of positive integers that are multiples of 5.

A: Solution

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: Show that one of the prime divisors of any three-digit composite number is less than 31 prime, or…

A: A composite integer a has a prime factor p only if p≤a that is if p≥a than there exist another prime…

Q: How many integers in the set {1,2,3,4, ..., 200} is divisible by at least one of the primes 2, 5 and…

A: Just use inclusion-exclusion principle for three sets.

Q: Use mathematical induction to show that given a set of n + 1 positive integers, none exceeding 2n,…

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Q: Construct four relatively prime integers a, b, c, d such that no three of them are relatively prime.

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Q: Assume that all four-digit integers (numbers from and including 1000 to and including 9999) are…

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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: Show that if n is an odd integer then []

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Q: compute the sum of all integers between 60 and 110 that are exactly divisible by 7

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Q: Any two integers are congruent modulo to: even integer odd integer Prime integer not a prime nor…

A: suppose n is a positive integer. Then two integers are said to be congruent modulo n if (a-b) is…

Q: Prove that the integer 359,240,758 cannot be a perfect quartic. (i.e. fourth power)

A: We have to prove that the Integer  359240758 can not be a perfect quartic

Q: Show that the set of a+bi Gaussian integers has the same strength as the set of natural numbers N.

A:

Q: Give a recursive definition for the set of all the positive integers that are less than 100 and not…

A: Given: Set of all positive integers that are less than 100. To find: Recursive definition for the…

Q: Prove that any integer greater than or equal to 48 can be represented as the sum of nonnegative…

A: Principle of mathematical induction: For each and every natural number n, mathematical induction is…

Q: Show that : If n is an integer then n is congruent to 0 modulo 4 or 1 modulo 4.

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Q: of

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Q: that the set of negative integers divisible by 3 is countable

A: Set of negetivee integers divisible by 3 isS=-3n, n∈ℕf: ℕ→S   defined asf(n) = -3n,To check…

Q: Show that in any collection of 7 or more natural numbers there are ALWAYS TWO numbers whose sum or…

A: In the given problem we have to show that in any collection of 7 or more natural numbers there are…

Q: Prove that any positive integer N is divisible by 11 if and only if the difference between the sum…

A: Consider the provided question, we need to prove that, Any positive integer N is divisible by 11 if…

Q: How many positive integers between 100 and 999 inclusive are divisible by 7?

A: The formula for term number n in an arithmetic series is . Here, a is the first term, n is the…

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

A: We have to prove that The given statement is for every integer n ≥ 2, if a set S has n elements.

Q: What is the set Z of all integers?

A: The set of all integers is, ℤ=........, -4, -3, -2, -1, 0, 1, 2, 3, 4, ........

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.