What is the converse of: an integer has divisors other than 1 and itself, if an integer is composite?

Q: 3. How many 4-digit positive integers are there that are even or contain no O's?

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Q: 2. Prove that 11" – 7" is divisible by 4 for any positive integer п.

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Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

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

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Q: An integer n > 2 is composite if and only if n doe

A: Given Statement: An integer n > 2 is composite if and only if n does not divide (n - 1)! + 1.…

Q: Let i be a positive integer and A, = {..., i – 6, i – 4, i – 2, i, i + 2, i + 4, i + 6, ... }. Then…

A: We have to find intersection of Ai

Q: Want a rigorous proof   Prove that The set of prime numbers has no largest element.

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Q: A natural number greater than 1 that has only itself and 1 as factors is called a/an number.

A: Given that  A natural number greater than 1 that has only itself and 1 as factors is called…

Q: If n is an integer, then 2n + 1 is odd

A: Given , n is an integer We have to show that 2n+ 1 is odd

Q: Use the Mathematical induction to verify that , for n any positive integer, 6" – 1 is divisible by 5

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Show by induction that if (n) is a positive integer, then ((n^3) - n) is divisible by 6.

A: Given statement is

Q: show that n(n^2-1)(n+2) is divisible by 4 for all integers n

A: To show that n(n^2-1)(n+2) is divisible by 4 for all integers n

Q: Show that an integer N is divisible by 9 if and only if the sum of its digits is divisible by 9

A: Let a=anan-1...a3a2a1 be an integer and S be the sum of the digits. Consider a,…

Q: Every natural number greater than 1 can be written as the products of prime numbers.

A: Prove that  every natural number greater than 1 can be expressed as product of primes or the number…

Q: If p + 2 is a prime number, prove that p² + 3 is divisible by 4.

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Q: Show by induction that 19"-8" is divisible by 11 for all n >0

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Q: How can I prove by induction that 2^n + 1 is divisible by three for all odd integers

A: To apply mathematical induction to show that 2^n+1 is divisible by 3 if n is odd

Q: Every two different prime numbers are relatively prime. False True

A: In this question we need to check the given statement is true or false

Q: How many integers from 1 to 1000: a) Are divisible by 7? b) Are divisible by 7 but not 11? e) Are…

A: a) From 1 to 1000 there are 1000 numbers. Now, 1000/7 = 142.86 So, there are 142 multiples between 1…

Q: If n is a natural number, the number 34n+4 – 43n+3 is divisible by: -

A: Given : To find : A positive number which can divide above expression

Q: If the product of two integers is 2'3*5?7!" and their great- est common divisor is 2°345, what is…

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Q: find all possible three-digit numbers that are of the form 75_ and are divisible by both 3 and 2

A: Since the number is divisible by 2 and 3, it must be divisible by 6.

Q: If z ≥ 2 is an integer, show that the following numbers are composite. n! + 2, n! + 3, n! + 4, ... ,…

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Q: If n = p,P2 ... P where the p, are distinct prime numbers, how many ways can n be written as a…

A: First let us formulate the problem in a different way. Look! we have to find the total number of…

Q: 7. Show that for every natural number n2 1, the integer ) + () +*(;) + - +) +22 +... +2" is…

A: Using binomial theorem:  n0+2n1+22n2....2nnn=n01n·20+n11n-1·21+n21n-2·22....nn1n-n·2n…

Q: (b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

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Q: 1)How many integers between 1000 and 2000 are divisible by at least one of the integers 3, 5 and 7?

A: Introduction: In mathematics, integers are the sum of positive and negative numbers. Integers are…

Q: Use indirect proof to show that if n is an integer and n^3 + 5 is odd, then n is even.

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Q: Prove or disprove: If an integer is divisible by 6 and by 10, then it is divisible by 60.

A: Answer

Q: Suppose “a” is an integer. If 5|2a, then 5|a.

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Q: Prove that 6-1 is divisible by 5 for any n 21.

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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: Prove that 9? −1 is divisible by 8,for all n ≥ 1

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Q: 4.The set of integers from 1 to 10 that is divisible by 3.

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Q: The positive integers that are divisors of 68 are

A: We will use multiple method to find all the divisor of the given number .

Q: Find out how many distinct three-digit numbers can be formed using al the digits of 1, 2, and 3. * O…

A: To find  Number of distinct three digit numbers can be formed using all the digits 1,2 and 3.

Q: Explain why it is convenient to write the expression for the number of possible permutations in…

A: It is efficient and time saving, imagine if you want to write 4! in its entire multiplication form…

Q: Show that 3-4" + 51 is divisible by 3 and 9 for all positive integers n.

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Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

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Q: If  n×k divisible by 4, then n is divisible by 4 or k is divisible by 4. Which of the following is…

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Q: If n is an even integer, then 3n° is an even integer.

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Q: How many negative divisors has the integer 90?

A: The divisors of 90 are all the numbers that we can evenly divide into 90.

Q: 4. Prove that given any integer for n, n3 + 2n will be divisible by 3.

A: We have to prove by induction.

Q: Assuming every integer is either even or odd, How would I show that for any integer, the difference…

A: Assume that every integer is either even or odd. Let the integer considered say x is even. Then by…

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: Show by induction that 21" – 8" is divisible by 13 for all n> 0

A: For the solution follow the next steps.

Q: Is it true that for any odd n the number 2" +1 is divisible by 3? Prove your answer.

A: We prove this by using PMI(Principle of Mathematical induction)

Q: Use proof by cases to show that if n is an integer then n^2 ≥ n.

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Q: Show by induction that 19" – 8" is divisible by 11 for all n >0

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Q: divisible

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