Find the number of ways a 5-digit positive integer can be made such that in each of them every digit is greater than the digit to its right.

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: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

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Q: Prove that any positive integer greater than or equal to 8 can be written as a sum of 3's and 5's.…

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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: 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: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: Prove that if n is a +ive integer,

A: Given that, M xddxnf(x);p = (-1)npnF(p) We can prove this by mathematical induction.

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

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

Q: Prove that 3"< n! whenever n is a positive integer greater than 6.

A: Here we are to prove 3n&lt;n ! whenever n is a positiveinteger greater than 6.To solve this problem…

Q: Prove that the product of any k consecutive positive integers is an integer multiple of k!.

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Q: rove that the negative of any even integer is even

A: Assume 2n to be an even integer. 2n = even integer -2n = negative of even integer  Let a = -2n be an…

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

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Q: of the first 150 positive integers?

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Q: Show that every positive integer can be written as the product of a square (possibly 1) and a…

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Q: Prove there are infinitely many prime numbers

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Q: a) How many 3-digit positive integers are there with odd digits?

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Q: Show that the cube for any integer is of the form 7k, or 7k ± 1

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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 the sum of any five consecutive positive integers is always a multiple of five.

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

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Q: Prove that there are infinitely many prime numbers

A: Let us assume that there exist finitely many prime numbers (Let n). Here, n is a positive integer.…

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

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Q: Find all integer numbers a for which

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Q: For each positive integer n, find with proof GCD(220° - 220, 220" + 218).

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Q: Discrete

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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: n + 2020 n + 2021 and LVn +1]

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Q: take any 11 integers. prove that there must be at least two of them that differ by an integer…

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Q: Prove that if n is an integer and n² is an even integer, then n is an even integer

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Q: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

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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: Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

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

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Q: Let S be the set of all (nonnegative) 4-digit integers. How many integers does S contain?

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Q: Prove that the number in between twin primes is always a multiple of 6.

A: A Twin prime are those numbers which are prime and having a difference of two ( 2 ) between the two…

Q: Prove or disprove: If three consecutive integers are multiplied together, and the second in order of…

A: We first look out some examples to understand the fact- choose three consecutive numbers be 1,2,3.…

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: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: There is no integer that has a reciprocal that is an integer.

A: As per the instructions, we will be answering only the first question. For (32), Notice that the…

Q: How many integers between 100 and 999 have distinct digits, and how many of those are even numbers?

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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: How large a subset of the integers from 1 to 50 must be chosen to guarantee that one of the numbers…

A: We have to determine How large a subset of integers from 1 to 50 must be chosen to guarantee that…

Q: Let be a prime number (integer). Show that p2 + 5 cannot be a prime number.

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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: Find all positive integers n such that both n + 2020 and n + 2021 integers. are LVn +1]

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Q: Show that |x^n| = |x| ^n for every positive integer n and every real number x.

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Q: show that for every positive integer n there is a prime whose decimal expansion contains n…

A: so we have to prove , for every positive integer n there is a prime whose decimal expansion contains…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.