3. Prove that least significant digit of the square of an even integer is either 0, 4, or 6.

Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

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Q: a. Find all pairs of integers such that one is a multiple of 7 and the other is a multiple of 11 and…

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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: Find a positive integer n such that 2n is a square and 3n is a cube.

A: Sol:- Let 2n be equal to x², where x is a positive integer. Then, n can be written as n = x²/2.…

Q: Prove that, all integers x and y, x^2-4y≠2  1a. Prove that all numbers of the form 1007, 10017,…

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

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Q: K = (perfect squares of integers} under addition? Is it closed or not

A: K is the perfect square of integerNow we find whether it is closed or not.

Q: Suppose that 75% of students attend lectures for a particular course and 20% of students fall the…

A: Conditional probability : Conditional probability is a measure  of the probability of an event…

Q: 10. Let x be a real number. Prove that at least one of the numbers /2 – x or /2 +x is irrational.

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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: How many integer solutions to x1 + x2 + x3 + x4 x₁ ≥ 4, x2 ≥ 2, x3 ≥ 5 and x4 ≥ 3? > = 31 are there…

A: Find change all inequality to grater then equal to 0. Then use the theorem

Q: 2. Prove that the difference of an odd integer and an even integer is odd.

A: Prove that the difference of an odd integer and an even integer is odd

Q: Prove that 6">5" + 9 for any integer n ≥ 2.

A: Mathematical induction is a technique used to prove statements about well-ordered sets of natural…

Q: s?

A: *Answer:

Q: Show that, for an integer n, if n^3 is odd, then n must be dd.

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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: (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: Prove that square of any integer leaves the remainder either 0 or 1 when divided by 4.

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Q: 2. Prove the statement: The sum of the squares of any two consecutive integers is odd.

A: I have assumed that n be any integer. Then i have found the sum of square of two consecutive…

Q: 3. Show that there exist 100 consecutive positive integers that are not perfect squares.

A: Here we have to show that there exists 100 consecutive positive integer that are not perfect square.…

Q: 2. Use Fundamental Theorem of Arithmetic to prove that the product of any two odd integers is an odd…

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Q: Prove by contradiction that a real number that is less than every positive real number cannot be…

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Q: show that no integer of the form n^3+1 is a prime other than 2

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