Prove that every integer greater than 27 can be written as 5a + 8b, where a, b e Z+.

Q: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

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

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

Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

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

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

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

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

Q: Prove that there exists an odd integer m such that every odd integer n with n greater than or equal…

A:

Q: Prove: there exist no integer a and b for which 21a + 30b = 1.

A: We can prove this by contradiction. Take the negation of the statement and check if it is true, if…

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

A:

Q: Prove that for any integer a, 3 divides

A: To prove that for any integer a, 3 divides (a-3)(a+4)(a+2).

Q: Prove that the cube of any integer has to be exactly one of these forms: 9k, 9k + 1, or 9k + 8 for…

A:

Q: Show that for any integer n, n² – 2 is not divisible by 3.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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

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

Q: Prove that there are no positive integers a and b such that a + b³ = 250.

A:

Q: Let a be a positive integer. Then Lcm(2a+1,2a+2)= *

A: The given numbers are 2a+1 and 2a+2. Clearly, the given numbers are two consecutive numbers.

Q: show that for eury A s integer nz3; A n- Am-2 +A2J. Mence determine ASO.

A: To show: That An=An-2+A2-I for n≥3 then find A50. Given: The matrix is A=100101010. Calculation:…

Q: For every positive integer n, prove that 7n – 3n is divisible by 4.

A:

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169.

A: Topic - principle of mathematical induction

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

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: For any non negative integer n , prove that an − bn is divisible by a−b,where a and b are any…

A: answer

Q: Every positive integer is the sum of at most two squares and a cube of nonnegative integers. O True…

A: consider 0  0=02+02+03 now  1 can be written as  1=02+02+13 now consider 2 2=12+02+13 now consider 3…

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

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

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

A: Given,  2+7+12+...+(5n-3)=12n(5n-1)

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

A:

Q: Prove that for all positive integers n, 4n-1 is divisible by 3.

A:

Q: b) Prove that () - 2) = n², where n is a positive integer.

A:

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: Prove that the statement is true for every positive integer n.3 is a factor of n3 + 2n

A: Consider the given statement, "3 is a factor of  n3+2n" for all positive integer n Suppose Pn…

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 statement is true for every positive integer n.

A: We have, Pn=a2n-1+b2n-1 Let us consider the statement Pn is true for ∀n∈ℕ. Therefore,…

Q: Show that there are infinitely many integers n such that 4n^2+1 is simultaneously divisible by 13…

A: We need to prove there are infinitely many integers n such that 4n^2+1 issimultaneously divisible by…

Q: Prove that the statements here are true for every positive integer n. a + (a + d) + (a + 2d) + · · ·…

A: To Determine: Prove that the statements here are true for every positive integer n. Given: we have…

Q: . Let a and b be integers. If 7 | (17a + 256), prove that 7 | (10a + 11b).

A:

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

A:

Q: prove that Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…

Q: 2. Show that if a is an integer other than 0, then a) 1 divides a h) a divides 0

A:

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

A:

Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

A:

Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

A:

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

Q: Prove that for a positive integer n we have that Bn+3 - 26n – 27 is divisible by 169.

A:

Q: Vn € Zt [obeb(n, n + 1) = 1]

A: l  let n,n+1 are two consecutive positive integers let  the greatest common divisor i.e…

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

A: We need to prove by mathematical induction that, 3 divides 17k+1-2*5k+3+11k+3 whenever k is a…

Q: Prove that if n is an integer, then 2n2 + n + 1 is not divisible by

A: The given problem is to prove the following statement using the possibile cases. We need to find the…

Q: a2 − b2is divisible by 5.

A: First observe that one's place of the square of a positive number can only one among 0,1,4,9,6, and…

Q: Prove that if n is an odd integer, then 3n+ 5 is an even integer.

A:

Q: For any non negative integer n , prove that an – bn is divisible by a-b,where a and b are any…

A: Use the binomial factorisation