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

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

A: Given, 3+9+15+..+(6n-3)=3n2

Q: Prove that 6 divides n^3 − n whenever n is a non-negative integer

A:

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

A:

Q: 23. Prove that for any integer a, 9/ (a² – 3). -

A: Prove that for any integer a, 9 ⫮ (a2 – 3).

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

A:

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

A:

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: Let a and b be integers. Show that if every integer that divides a also divides b, then a | b. (a…

A:

Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

A:

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

A: To prove: 1+4+7+...+3n-2=n3n-12

Q: Prove that if a > 2 and b are integers, then a † b or a { (b+1).

A: Given that a≥2 and b are integers. We will prove it by contradiction. Let a|b and a|(b+1) then there…

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

A: Given that, 2 is a factor of n2+n   We have to prove this statement is true for every positive…

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: Prove that 2n+1 +(-1)" 3-2 whenever n is a nonnegative integer.

A:

Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

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

A:

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

A: We know that the sum of a geometric series of the form: a+ar+ar2+ar3+...........+arn-1 with the…

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

A:

Q: Prove that the product of two odd numbers is odd.

A: We first define a general odd number. Lets say that n is an integer. When we multiply any number by…

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.…

Q: Show that is a and b are positive integers such that (a | b) and (b | a), then a = b

A:

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

A:

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

A:

Q: Prove or disapprove that if a|bc, where a,b,c are positive integers and a≠0, then a|b or a|c

A:

Q: If a and b are distinct integers prove that a-b is a factor of an -b" whenever n is positive integer

A: if a and b are distinct integers prove that a-b is a factor of a^n - b^n whenever n is positive…

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 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: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

A:

Q: prove that if a and b are both odd integers, then ab+1 is an even integer

A:

Q: Prove that if ni is a positive integer, a± 0

A:

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

A: We use mathematical induction. Let P(n) be "3 divides n3+2n" For n=1 n3+2n=(1)3+2(1)=3 We then note…

Q: Prove that the difference between two integers is odd if and only if exactly one of the integers is…

A: Suppose x be any integer. If x is odd then x=2m+1 for some integer m. If x is even then x=2m for…

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

A:

Q: 2. Determine all values of x - y when x and y are integers such that x = 360 + y².

A: The given equation is. x2=360+y2x2-y2=360x+yx-y=360 Since it is given that x and y are integers, so…

Q: Prove that 3, 5, and 7 are the only three consecutive odd integersthat are prime.

A:

Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: Prove or disprove: If a and b are odd integers, then 4 | (a – b) or 4 | (a+b). -

A:

Q: Prove that it is impossible to choose three distinct integers (a – b)|(b – c), (b – c)|(c – a), (c –…

A:

Q: Prove 8 divides 34h+1+52h+1 whenever h is a positive integer

A: We need to prove that 8 divides 34h+1+52h+1 whenever h is a positive integer by Mathematical…

Q: Prove that n2 +1 > 2? when n is a positive integer with 1<n<4.

A:

Q: 4. Prove that if m and n are positive integers and x is a real number, then

A: We have to prove here that the expression here follows both commutative and associative property. If…

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: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

Q: Prove that if the product of two integers ab is odd, then a2 + b2 is even

A:

Q: Prove that if a, b and c are positive integers such that a divides b and b divides c, then a divides…

A: It is given that a,b and c are three positive integers. It is also given that a divides b. Then it…

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y? is odd.

A: We have to solve given problem:

Statements and Logic

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