3. Use induction to prove that 15"+2 + 162n+1 is divisible by 241 for every nonnegative integer п.

Q: Use the Principle of Mathematical Induction to show that for every neZ,, 52n-1 +1 is divisible by 6.

A: To prove that for every n∈ℤ+, 52n-1+1 is divisible by 6 using the Principle of Mathematical…

Q: 4) Prove that the product of any three consecutive integers is divisible by 6.

A: We have to show that product of any three consecutive integers is divisible by 6

Q: Use induction to prove that the product of any three consecutive positive integers is divisible by…

A:

Q: Prove, using the definition of odd, that for all integers r, that 6r+5 is odd.  b) true or false:…

A:

Q: Use Wilson's Theorem to verify that 4(29!) + 5! is divisible by 31.

A: The given number is 4 (29)! + 5! . Our aim is to verify that the given number is divisible by 31…

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 by induction that 32n 1 is divisible by 8 for Vn 20

A:

Q: Prove that 8 | (54" +7) for every positive integer n.

A:

Q: (3) Prove that if a cube of a number is divisible by 17, then the number is divisible by 17.

A: We have to prove that if cube of a any number is divisible by 17 then the number itself is divisible…

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: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: Prove this: -n is a integer; If 6| n then 18| n^2 +3n

A:

Q: what two consective odd integers have a product of 63?

A: Find two odd integers.

Q: 512|8.5n + 6.32n -80n -14

A: Given: 512|8*5n + 6*32n -80n -14

Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

A:

Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Use induction to show that 12 -7 is divisible by 5.

A:

Q: Prove {12a + 4b : a and b are integers} = {4c : c is an integer}

A:

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: If p2q25 is a prime number, prove that p-q² is divisible by 8.

A: Given, p≥q≥5 are prime numbers, Now, p2-q2=p+qp-q. Since, p and q are prime which are grater than…

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 3" + 1 is divisible by 4 whenever n is an odd positive integer.

A:

Q: Use Mathematical Induction to show that 52n -1is divisible by 8,   ∀ ∈ n N.

A: Use Mathematical Induction to show that 52n -1is divisible by 8,   ∀ ∈ n N.

Q: Use mathematical induction to show that 3 divides n3 + 2n whenever n is a nonnegative integer.

A:

Q: Use induction to prove that n^3 + 2n is divisible by three for all integers n greater than or equal…

A: Given information:  n3+2n is divisible by three for all integers n greater than or equal to zero.…