Compute the sum of the first 60 positive integers that are exactly divisible by 4.

Q: Determine whether there are infinitely many primes that have a remainder of 12 when divided by 25.…

A: Given question is Determine whether there are infinitely many primes that have a remainder of 12…

Q: 2. Prove that " The sum of two odd numbers is even: using direct proof.

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Q: Prove that Master Cheng has won precisely 21 times in a row during some consecutive days.

A: Please find the solution

Q: Show that the sum of all  odd  integers between 1 and 1000 which are divisible by 3 is 83667

A:

Q: Suppose that 54% of people own dogs. If you pick two people at random, what is the that they both…

A: Given: P(Dog) = 0.54 The probability that they both own a dog can be calculated as: P(Both own a…

Q: Show that for positive integer n: (2n)! VT. 22n . n! Г 2

A:

Q: 4) Prove that the product of any three consecutive integers is divisible by 6.

A: We have to show that product of any three consecutive integers is divisible by 6

Q: Show that there are infinitely many primes of the form 4k + 3.

A: We have to show that there are  infinitely many primes of the form 4k+3.

Q: Find S, the sum of first 15 Natural numbers 15

A: We have to find sum of first 15 Natural numbers.   S15? Solution :  As we know that the sum of first…

Q: Prove the formula for the sum of the first positive integers:

A:

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

A:

Q: Prove that the sum of any five consecutive positive integers is always a multiple of five.

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Q: The sum of all positive integers less than 300 which are divisible by 3.

A:

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: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

A:

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: Find the sum of the positive even integers up to and including 50.

A:

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

A:

Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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

A:

Q: The sum of all positive integers less than 300 which are not divisible by 3.

A: Sum of n positive integer is given by, Sn=nn+12

Q: Compute P5,2. (Enter an exact number.)

A: Obtain the value of P5,2.  The value of P5,2 is obtained below:   Permutation: A permutation is an…

Q: Compute the sum of the first 50 positive integers that are exactly divisible by 5.

A: To determine sum of the first 50 positive integers that are exactly divisible by 5.

Q: How many combinations are there to pick r objects from 2n objects numbered from 1 to 2n with n > 2…

A:

Q: If n > 1 is an integer not of the form 6k + 3, prove that n + 2" is composite. [Hint: Show that…

A: Congruence modulo relation on integers: Let a, b∈Z and let m be a fixed positive integer. Then we…

Q: Compute P5,4. (Enter an exact number.)

A: Obtain the value of P5,4.  The value of P5,4 is obtained below:   Permutation: A permutation is an…

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: 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: find the sum of all even intergers less than 1000 that are exactly divisible by 7

A:

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

A:

Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

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

A:

Q: The sum of all positive integers less than 400 which are divisible by 5.

A: Positive integers less than 400 which are divisible by 5 are⇒n=79 5,10,15,20,...,395 The above…

Q: Use a direct proof to show that the sum of two odd integers(k and l) is even.

A: The number n is even if it is expressed as  n=2m , where m is an integer. The number n is odd if it…

Q: Find all three-digit numbers with different digits such that those three-digit numbers are divisible…

A:

Q: Compute P9,2. (Enter an exact number.)

A:

Q: How many 6-permutations of [15] have their digits in increasing order?

A: Permutation is a technique to obtain the possible arrangements of listed items in a linear order.…

Q: sum of two even integers is even

A:

Q: Prove by contradiction that 2020 cannot be written as a sum of 3 odd integers.

A: The objective is to prove  that 2020 cannot be written as a sum of 3 odd integers by the…

Q: Prove that no integer congruent to 3 modulo 4 can be written as the sum of two squares.

A:

Q: How many sequences of 6 positive digits (1-9) contain at least one digit that appears twice?

A: We have to determine the total sequences of 6 positive digits (1-9) that contain at least one digit…

Q: Don't copy from google or other sites  How many bit strings of length 11 contain either five…

A: Given, the strings of length 11 contain either five consecutive 0s or five consecutive 1s.

Q: Show that if seven integers are selected from the first 10 positive integers, there must be at least…

A: We can solve this question by pigeonhole principle

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: For n > 3, prove that the numbers ±5, ±52, · reduced residue system modulo 2". form a

A: A subset R of the integers is called reduced a residue system modulo n if (i) gcdr,n =1 for each r…