Show that given any set of 5 natural numbers, there are 3 numbers whose sum is divisible by 3.

Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

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Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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Q: How many 3-digit positive integers exist that when divided by 7 leave a remainder of 57

A: 3 digit positive integers start from 100 and ends with 999. Least positive 3 digit integer which…

Q: Prove the statement. "For any integer n, nº +4 is not divisible by 8." by Contradiction.

A: Given: n be any integer. To prove: n3+4 is not divisible by 8.

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

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Q: Determine the number of natural numbers less than 700 that can be written using the digtits 5,6,7,9?

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Q: Let ab5c be all four-digit multiples of number 36 with different digits. If someone found 7 numbers…

A: Let ab5c be all four digit multiples number 36 with different digits.   As 36= 4x9, so the number…

Q: of the first 150 positive integers?

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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: The set of numbers whose decimal representations are neither terminating nor repeating is called the…

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Q: Prove that the sum of any three consecutive numbers is divisible by 3

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Q: -, 1+2+2 '+.+ 2 -1 for all natural numbers n.

A: We will use the principle of mathematical induction to prove the result.

Q: Prove that 2^10 + 5^12 is a composite number.

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Q: Are the irrational numbers closed under multiplication?

A: Use closure property.

Q: Prove that an odd number multiplied by an odd number is odd.

A: The objective is to prove that an odd number multiplied by an odd number is odd.

Q: Set C is the set of positive two-digit even numbers less than 56 that are divisible by 5.

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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: How many pairs of two distinct integers chosen from the set {1, 2, 3,..., 101} have a sum that is…

A: Since you have posted multiple questions and we can answer one question. So answer provided for the…

Q: Show that the set of natural numbers, N is not compact

A: To prove that set of natural nubers is not compact , first we give Heine-Borel theorem and…

Q: Define an odd number and show that the product of odd even numbers is also even.

A: We need to first define an odd number. And layer prove product of an odd and an even number is…

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: Let A be the set of all positive integers whose last two digits form a number that is evenly…

A: We have the given condition as   Let us consider A be the set of all positive integers whose last…

Q: Write the set of natural numbers greater than 16 using the most concise method.

A: We need to write the set of natural numbers greater than 16 using most concise method. We know that…

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: Show that n-lis divisible by 8, if n is an odd positive integer

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Q: Show that there are infinitely many prime numbers.  0 /∈ N. (zero is not an element of natural…

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Q: Prove that the product of an even number and an odd number is even.

A: Here, objective is to prove that the product of an even number & an odd number is even

Q: show that every integer greater than 11 is the sum of two composite integers

A: We need to prove that  every integer greater than 11 is the sum of two composite integers.

Q: (a) Suppose that there are k distinct positive integers that divide an odd positive integer n.…

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Q: Prove that if the sum of the digits is subtracted from a natural number, then the result is…

A: let us assume the number to be three digit number xyz therefore, xyz=100x+10y+z sum of digits will…

Q: identify all natural numbers having exactly 28 divisors.

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Q: What is the set N of all natural numbers?

A: Set of all integers: ℤ={.......,-3,-2,-1,0,1,2,3.......} Set of all positive integers…

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

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Q: 12 belongs to the set of natural numbers

A: The natural numbers (i.e. counting numbers) are numbers which are used for counting and ordering.…

Q: Determine all natural numbers divisible by 8 whose sum of digits is less than 10 and the product of…

A: Permutation and combination concepts will be used to filter out all the 3 scenarios in getting…

Q: Prove that 8 divides 9" – 1 whenever n is a natural number. For each natural number n, let P(n) be…

A: To prove that  8 divides 9^n-1 whenever n is a natural

Q: Show that the set of odd numbers is countable.

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Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

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Q: How many integers from 1 to 99,999 contain exactly two consecutive 1s?

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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: Two numbers a and b are chosen from the set 1,2,.., 3n In how many ways can these integers be…

A: In the question we have to find total no. of cases by which a2-b2 is divisible by 3.

Q: Find the sum of all  natural numbers between 250 and 1000 which are exactly divisible

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Q: The product of all the natural numbers from 1 to n is written as n! it is read as "n factorial" true…

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Q: How many corresponding integers are made up of 1443 digits taken from (1-2-3-4-5-6) provided that no…

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Q: How many four-digit positive integers both start and end in even digits:

A: We need to find the number of positive integers both start and end in even digits.