The sum of any two integers of the from 4k+ is always: a) prime b) zero c) Even

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: Use a direct proof to show that the sum of two odd integers is even.

A: Types of integers : Odd integers : The integer which is not divided by 2 . It has the form 2k+1…

Q: plz solve q4

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Q: Find all primes that are the difference of the fourth powers of two integers.

A: Let the two integers are x and y and n is prime so we have  n=(x4-y4)       (suppose x>y)…

Q: he sum of any two integers of the from 4k + 1isalways: COdd prime Zero even

A: The sum of two consecutive integers can always be written as 2n+1, but it can be written as 4n+1…

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: Faces of a polyhedron are painted into two colors, so that no two faces sharing an edge have the…

A: Consider the provided question, We need to prove that the number of edges of all faces of polyhedron…

Q: How many positive integers less than 850 are multiples of 8, 3, or 5? Use the Principle of…

A: Using the Principle of inclusion/Exclusion, the formula for A∪B∪C can be written as:…

Q: Prove that 5 does not divide 5n+3 for any integer n.

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Q: The diagram below shows ten circles joined by five lines with three circles Place all the…

A: Requirements: Numbers should range from 1 to 10.   Every line has three circles.   3. The amount for…

Q: b. Prove that the integer 359,240,758 cannot be a perfect quartic. (i.e. fourth power) that 250 758…

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Q: Show

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Q: Show that if m is an integer then 3 divides m3−m.

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

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Q: Show that if any eight positive integers are chosen, two of them will have the same remainder when…

A: Pigeonhole Principle- If n items are assigned to m containers, and m<n,then at least one…

Q: -ん 2 kodd

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Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

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Q: . Let m be any odd whole number. Then m is a divisor of the sum of any m consecutive whole numbers.…

A: I have supposed that m is any odd whole number and then I have taken another arbitrary whole number…

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

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

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Q: 32. Prove that 3 divides n + 2n whenever n is a positive integer.

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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: what two consective odd integers have a product of 63?

A: Find two odd integers.

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

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Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.

Q: Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it…

A: (a)The fill up for the statement to rewrite the given statement: Is there an integer n such that n…

Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: 2. Find all Pythagorcan Triples (not just primitive ones), containing the integer 15.

A: The Pythagorean triples are positive integers a,b,c such that a2+b2=c2. We can say that the…

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

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Q: s) Express the infinite contimod fraction [2, 1,5 quadratic irrational- as a ity.

A: We need to express the infinite continued fraction 2, 1, 5¯ as a quadratic irrationality. The…

Q: Prove that 4 does not divide n2 +2 for any integer n.

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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: 4. Find the sum of positive divisor of 5,020 and also find the pair of this number so that they can…

A: We have to find the sum of all divisor of 5020 and then the amicable pair of the number.

Q: How many zero divisors are in Z,, where n = 32547?

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Q: Can you think of the smallest number N such that ot can be expressed as a product of the two factors…

A: We have to find out the number N that can be expressed as a product of the two factors which are…

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: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

Q: 1. Show that if seven integers are selected from the first 10 positive integers, there must be at…

A: We can solve the following question by pigeonhole principle

Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

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Q: Consider an enclosure consisting of eight surfaces. How many view factors does this geometry…

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Q: Exercise 5. Prove that given positive integers a and b, there is a positive integer d such that • d…

A: Given two positive integers a and b, let us choose d = gcd(a, b) [greatest common divisor] Then…

Q: 5!, called 5_______ , is the product of all positive integers from_________ down through________ .…

A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: Find the sum of all natural numbers n such that 2021-n/99 is a natural number ?

A:

Q: Prove by induction that the product of the first n even numbers is equal to n!2" for all positive…

A: We have to prove by induction that the product of first n even numbersis equal to n!2n for all…

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…

Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…

Q: What 2 nonnegative numbers with a sum of 23 have the largest possible product?

A: Given information: The two non negative numbers with a sum of 23.  Calculation: Consider x and y are…

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.