Show that every positive integer can be written as the product of a square (possibly 1) and a square-free integer.  A square-free integer is an integer that is not divisible by any perfect squares other than 1.

Q: Prove that (-1ti3)" + (-1-is3)" has either the value 2n+' or the value -2" if r is any ). integer…

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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 by cases that 25x² + 35x is an even integer whenever 5x-7 is an integer. Note here that the…

A: Apply the condition for 5x-7 to be an integer case by case.

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: 2. Prove that 11" – 7" is divisible by 4 for any positive integer п.

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

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Q: 1) Show that the square of an odd integer is odd. 2) Show that the product of two odd integers is…

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Q: Show that if a positive integer is odd, then it's square is odd.

A: To show that if a positive integer is odd then its square is odd

Q: 2. Prove that the product of any two odd integers is an odd integer.

A: Let x and y be two odd numbers. If a and b are odd, then a and b can be written as: x=2m+1 , y=2n+1…

Q: Let n be an integer. Prove that if 2n +4 is divisible by 4, then n is even.

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Q: Prove the square of any odd integer is expressible in the form 8n+1 with n ∈Z.

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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: Use a direct proof to show that every odd integer is a difference of two squares. (For example: 1 =…

A: first we take a odd integer then we write in 2y+1 form then we transfer to difference of two…

Q: Find the number of ways a 5-digit positive integer can be made such that in each of them every digit…

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Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: Determine the smallest subring of Q that contains 2/3.

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Q: 75. Prove that the sum of an even integer and an odd integeris odd.

A: sum of an even integer and an odd integer is odd.

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

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Q: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

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

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Q: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

Q: Define: "A positive integer alb). a is divisor or factor of integer b" (Notation: an

A: We Define: "A positive integer a is divisor or factor of integer b "(Notation: a|b).

Q: In the 1989, Journal of Number Theory paper, N. Tzanakis and B. de Wegner showed that the largest of…

A: We have to factories the equation  x3+3x2+2x-258,474,216=0

Q: Use the Division Algorithm to prove that (a) The square of any integer is either of the form 3k or…

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Q: factorization

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Q: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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Q: Find all integer numbers a for which

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Q: Prove that there are 1000 consecutive positive integers that are not perfect squares.

A: Here I give an example. That proves the hypothesis.

Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

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Q: Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

Q: Prove that the statement is true for every positive integer n.

A: we have to prove that the statement is true for every positive integer n. 4 is a factor of 5n-1

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: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

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: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Prove that for all primes p2 5, p + 1 cannot be a perfect square. Recall that is a perfect square if…

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Q: Prove that the fourth power of any integer has one of the forms 5k or5k + 1, where k is an integer.

A: We need to prove that any integer has one of the forms 5k or 5k + 1, where k is an integer.Let p be…

Q: Suppose that all the entries in A are integers and det A = 1. Explain why all the entries in A–1 are…

A: Inverse of a matrix A is calculated by,

Q: Prove that the difference between the square of any odd integer and the integer itself is always an…

A: An odd integer is of the form 2n+1; n ∊ ℤ. Consider the difference between the square of an odd…

Q: Prove that for every positive integer n, there are n consecutive composite integers.

A: This question is related to Descrete Mathematics

Q: The positive integers that are divisors of 68 are

A: We will use multiple method to find all the divisor of the given number .

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: Find all positive integers n such that the sum of all positive divisors of n is odd.

A: Given the data

Q: (a) Show that 1++++ 1 21+, 2n-1 2n whenever n is a nonnegative integer.

A: We need to show that 1+12 + 13 + ..... + 12n-1 + 12n ≥ 1+n2We will show this by mathematical…

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: Determine whether the statement is true or false. Justify your answer with a proof counterexample,…

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Q: Show that any integer of the form 6k + 5 is also of the form 3j + 2, but not conversely.

A: 2. We can write 6k+5 = 6k+3+2=3(2k+1)+2 lets write 2k+1=j hence 6k+5=3j+2

Statements and Logic

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