Every integer n22 either is a prime or can be expressed of a product of primes. This statement is known as:

Q: Let n denote an integer. Prove that 5|n and 7|n if and only if 35|n.

A:

Q: Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

A: n is an integer. To show: 2n3 + 4 is divisible by 4 then n is even.

Q: If n is an integer such that (m² – 1) is odd. Then n is also an odd integer. The following statement…

A: Explanation of the solution is given below....

Q: prove the following statement: Let n be an integer. If a and b are integers such that a is divisible…

A:

Q: Suppose that the number 'a' has a property that for every natural number n, a ≤ 1/n. Prove that a ≤…

A:

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

A: To show  33n+3 - 26n - 27 is divisible by 169.

Q: Prove that if n is an odd integer then 7n+5 is even.

A: Topic:- algebra  Note :- if more than one question uploaded than only first question need to be…

Q: Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp…

A:

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

A:

Q: ! Prove the following statements. (a) If k is a positive integer, then 3k+2 and k+1 are relatively…

A:

Q: Prove that for any integer n, we have n - 2 is not divisible by 4.

A:

Q: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

Q: For every positive integer a, find a composite number n such that n|a^n - a.

A:

Q: Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? + 1 is divisible by 10.

A: We Shall solve one question only un per the company guidelines as you have asked more than one Que.…

Q: For any integer ? show that the pair of integers 7? + 9 and 4? + 5 are relatively prime.

A:

Q: .If p 5 is an odd prime, prove that either p? – 1 or p² + 1 is divisible by 10.

A:

Q: Prove by using a direct proof that if n is an odd number, then n2 is also odd.

A:

Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: a. Show that for every non-negative integers m, n, N = 7.5" has exactly (m + 1)(n+1) different…

A: (a) Given that N=7m·5n To find : Number of positive divisors. Please post the multiple questions…

Q: Use induction to prove that for every integer 15"+2 + 162n+1 is divisible by 241. n > 0,

A: To use mathematical induction to prove the given statement

Q: Before arriving at his famous conjecture, Goldbach had conjectured that every odd positive integer…

A: Goldbach conjecture says that every integer greater than 5 is the sum of three primes.

Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

Q: . Prove that 3| (52n – 1) for every integer n 20.

A: Solution:  To show :   3|(52n-1)     , for every integer n≥0

Q: If a is any integer, is a(a + 1) even or odd?

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: How to solve this?

A: According to the given information, let p be a prime number such that p>2. It is required to…

Q: Give a direct proof of the following statement. If x is an even integer, then x² – 6x + 5 is odd. |

A: Remark: We know that addition or subtraction of an even and an odd integer is always an odd integer.…

Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

A:

Q: integers a, b, and c with a≠0, if a∣b,  then a∣(bc). I need a know show table and a proof

A:

Q: Show by induction that for each positive integer k, 1?+ 2² + 3² + . . · +k² = }k(k+ 1)(2k + 1).

A: NOTE: Refresh your page if you can't see any equations. . check for k=1 the statement is true…

Q: Let p be a prime number such that p> 2. Use the proof by contradiction to show that at least one of…

A: See the solution attentively I think you will understand easily.

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

A: This question is related to Descrete Mathematics

Q: 9. Ifn is a squarc-frce integer, prove that r(n) = 2" , where r is the number of prime divisors of…

A:

Q: . Let n be an integer. Prove that n is odd if and only if 3n? – 7 is even.

A:

Q: (a) Express each of the integers 4278 and 71929 as a product of primes. (b) What is gcd(4278,…

A:

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

A:

Q: Suppose n is an integer. Prove that if n is even, then n²+3n+5 is odd.

A:

Q: For every positive integer a, find a composite number n such that n|a" – a.

A:

Q: Give an example of a prime p of each of the forms: 6k + 1 6k + 5.

A: We need to give an example of a prime p of each of the given forms. Let us take k=1. So, we get…

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

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: Prove by Induction that for every natural number п, 3h-1 is a multiple of 2?

A:

Q: Show that if 2m +1 is an odd prime, then m 2" for some nonnegative integer n.

A: We solve this question by contradiction

Q: Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

A:

Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

A:

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

A:

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

A: 13+23+33+---+n3=nn+122 Let us use the Principle of Mathematical Induction to prove the given…

Q: Prove that for any integer n, if n³ is odd then n is odd.

A:

Q: is a prime number and that n is an integer such that p|n^2 prove that it follows p|n

A:

Statements and Logic

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