For each natural number n, if 3 does not divide (n² + 2), then n is not a prime number or n = 3.

Q: d) How many 4 digit numbers can you make from the elements of A without repetition contain the…

A: We shall solve first question only as per answering guidelines. If you want others to be solved…

Q: Find the greatest natural number n such that n + 10 divides nº + 100.

A:

Q: Each letter of the alphabet is assigned an​ integer, starting with A=​0, B​=1, and so on. The…

A: Solution: The objective is to find what​ two-letter word is represented by the digits ​02.

Q: n(nº +5) is divisible by 6, for each integer n 0.

A:

Q: How many three-digit numbers can be formed under each condition? (a) None of the digits can be 9.…

A: Here 3-digits numbers can be formed. a) 9 not be used b) lead dgits 0 to 4

Q: For each natural number n, the expression 2* – 7n - 1 is divisible by

A: Concept: Mathematical induction: Mathematical induction is a mechanism for proving a statement,…

Q: Every natural number greater than 1 can be written as the products of prime numbers.

A: Prove that  every natural number greater than 1 can be expressed as product of primes or the number…

Q: If n is an integer and n 2 1, then the expression 4" + 4-1.3+42.3+ 4-.3+4".3++4 3+ 3 is equal to

A: (c) is correct We have to find the value of 4n+4n-1·3+4n-2·3+4n-3·3+4n-4·3+⋯+4·3+3

Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is P(0)…

A: Replace the values of n by given values

Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is P(0)…

A: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is…

Q: How many numbers between 1 and 10000 have the digits 1,3, and 5 such that each digit appears exactly…

A: Here Numbers between 1 and 10,000 have digits 1,3,5 and each appears exactly once only

Q: Do you thinkthat there are infinitely many primes of the n -1

A: there are infinitely many primes of the n -1 Explain the above statement

Q: If k is a positive integer, find the (n!)k + 6 Σ ((k + 6)n)! n = 0 R =

A:

Q: Find all four-digit numbers abba with the property aa . 10b= abba if a is not equal to b.

A: We know if abcd is a four - digit number then, abcd = 1000a+100b+10c+d

Q: If n is a natural number, the number 34n+4 – 43n+3 is divisible by: -

A: Given : To find : A positive number which can divide above expression

Q: If n is an even integer then 7- n3 ) is an odd integer. ) is an even integer. is sometimes even and…

A:

Q: Show that for all integers n:  ceil(n) = -floor(-n)

A:

Q: Any n consecutive integers will include a number that is divisible by n.

A:

Q: For each natural number n, the expression 23n – 7n – 1 is divisible by

A: The binomial theorem is given by  (x+a)n=C0n xn a0+C1n xn-1 a1+...+Cnn x0 an…

Q: If z ≥ 2 is an integer, show that the following numbers are composite. n! + 2, n! + 3, n! + 4, ... ,…

A:

Q: If n = p,P2 ... P where the p, are distinct prime numbers, how many ways can n be written as a…

A: First let us formulate the problem in a different way. Look! we have to find the total number of…

Q: (b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

A:

Q: Every odd prime number is greater than 2.

A: Given: "Every odd prime number is greater than 2." To find: Statement in the form "If p, then q"…

Q: 5. Given a real number x and a positive integer k, how many multiplications are used to find kx by…

A: Given,A real number x and positive integer kTo find: Total number of multiplications used with the…

Q: For each natural number n A, = (-→ 3+n U-1 Find UneN An and NnEN An

A:

Q: (a) Suppose that there are k distinct positive integers that divide an odd positive integer n.…

A:

Q: Suppose “a” is an integer. If 5|2a, then 5|a.

A:

Q: If n is a product of twin primes, then a) Ø(n)o(n) = (n – 1)(n – 3) b) ø(n)o(n) = (n + 1)(n – 3) %3D…

A: If n is a product of twin primes, then a)  ϕ(n)σ(n) = (n+1)(n-3) b)  ϕ(n)σ(n) = (n+1)(n-3) c)…

Q: 8n? a prime number for every n = 1, 2, 3, . ...

A:

Q: For every natural number n, one of n-1, n, or n+1 is a multiple of 3

A: The following two steps are used to prove a statement P(n) by the principle of mathematical…

Q: all > 4, 2" – 1 is not a prime number.

A: Statement is false

Q: If an integer n is odd, then n2 + 2n +1 is even. Select one: True O False

A: Given n is odd Now n2 + 2n + 1 = (n+1)2

Q: Find all positive integers n for which n1º +1 is divisible by 10.

A:

Q: Let m and n be positive integers.

A: Given, m and n be positive integers.  To prove that mℤ⊕nℤ is an ideal of ℤ⊕ℤ and that…

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: The number of primes of the form n³ – 1 is 1) 1 2) 2 3) Infinite

A: To choose correct option

Q: fm and n are odd integers, then so is m · n.

A: Given query is to find statement true or false.

Q: If n is an even integer, then 3n° is an even integer.

A:

Q: Let p and q be prime numbers. (1) If p is not equal to q, how many generators does Zpq have? (2) If…

A: Given p and q be primes.

Q: Find n, if 3n+18=0

A: Linear equation question

Q: For any natural number n, n²+1 is always odd. True False

A:

Q: Find all values of primes p for which (¹¹) = 1 Р

A:

Q: Every prime number is odd. True False

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Q: Every natural number n > 2 cam be writtenn = PiP2 Pr as the ... product of distinct prime numbers…

A:

Q: The product of all the natural numbers from 1 to n is written as n! it is read as "n factorial" true…

A:

Q: (c) For each integer n, if n is odd, then 8 divides (n4 + 4n² + 11).

A:

Q: if n zo is an integer, then n!=n(n-)3.21 ...

A:

Statements and Logic

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