Prove that (5n)!÷40^n×n! is natural number for every natural number n​

Q: TION 9 Prove by Induction that for every natural number n, 3"–1 is a multiple of 2? Attach File…

A: We prove by induction

Q: Show that there are infinitely many primes.

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Q: Prove that the hundreds digit of n20 is even.

A: To Prove that the hundreds digit of n20 is even.

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: the product of any odd number and any even number is even.

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Q: Use Fermat's little theorem to find the least residue of 8^70 modulo 23

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

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: What is the smallest integer that is a quadratic residue for all prime moduli p > 2?

A: We know if there is an integer 0<x<p, where p is a prime, such that x2≡qmod p, then q is said…

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: 3. Show that the weighted arithmetic mean of first n natural numbers whose weights are equal to the…

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

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Q: Starting with the integer 72021, we remove the leftmost digit and add it to the remaining integer.…

A: Observation 1: After each step, the difference of the numbers before and after is divisible by 9.…

Q: 2"-1 for positive integers n.

A:

Q: Prove: in L number psiz 12, for every prime poiz, [p] = [1], [5) , [7) , or [11]

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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: Every even integer greater than 2 is sum of two primes? Give your views on this?

A: We know that 2 is even prime. And all other primes are odd.

Q: Prove that all generators of Z20 are prime numbers. Please provide complete solutions. Thank you.

A: The statement is incorrect as 9 is generator of Z20, but 9 is not prime number.

Q: 1. How many positive integers of 5-digit-number can be formed from the prime numbers starting at 1…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Prove that for all divides natural numbers nt 7^-2^.

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: how to prove √6 is irrational

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Q: Show that √17 is Irrational number

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Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

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Q: "Every positive integer can be written as the sum of the square of 3 positive integers." prove…

A: Given: "Every positive integer can be written as the sum of the square of 3 positive integers." To…

Q: (4) Prove that there is no smallest positive real number.

A: I have to prove that there is no smallest positive real number . Suppose that there is a smallest…

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

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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: All Fermat's number is clearly odd numbers

A: Fermat number.

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: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: Find the smallest positive integer for which x mod 3=2 and x mod 4=3   What is the next smallest…

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Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

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Q: prove its true by mathematical induction

A: The mathematical induction technique is used to prove the validity of the statement for every…

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: Show that — ф+1 n=0 ||

A: Given φ is the golden ration satisfying 1φ=φ-1. To prove ∑n=0∞1φn=φ+1 It is known that φ=1+52.

Q: Show that the product of two numbers of the form 4n + 1 is still of that form. Hence show that there…

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Q: If x is an odd integer then 5x – 7 is an even integer. (use direct proof)

A: Given that “x” is an odd integer then “5x-7” is an even integer.

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: (5n)! Prove that 40"-n! is natural number for every natural number n.

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Q: Use binomial theorum to expand (1 + 1/x)^7

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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.

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

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Q: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

Q: Give a proof by contradiction that there does not exist an integer which is both even and odd.…

A: Assume an integer x which is both odd and even.

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: algorithm to determine the decimal number answer to 2893 ÷ 6 to the hundredths place. b. Interpret…

A: Solution of the problem as follows