Every natural number has a successor every number except 1 has a predecessor

Q: TION 9 Prove by Induction that for every natural number n, 3"–1 is a multiple of 2? Attach File…

A: We prove by induction

Q: Any number n is congruent to the sum of its digits modulo 4. Select one: True False

A:

Q: Use mathematical induction to prove that for all natural numbers n, 3^n- 1 is an even number.

A: For n=1 , 31-1= 3-1=2 , this is an even number Let for n=m, 3m-1 is an even number.  Assume that…

Q: Prove that for each natural number n, 5 divides n^4 − 1 if and only if 5 does not divide n. please…

A: Consider the given information. We have to prove that for each natural number n, 5 divides n^4 − 1…

Q: Find all primes that are the difference of the fourth powers of two integers.

A: Let the two integers are x and y and n is prime so we have  n=(x4-y4)       (suppose x>y)…

Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

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

A: Every integer n≥2 is a prime or can be expressed as product of primes. This statement is Fundamental…

Q: For each nonnegative integer i, what is the least residue modulo 9 or 10i?

A:

Q: Find the primitive roots (mod 17).

A:

Q: Find the number of ways a 5-digit positive integer can be made such that in each of them every digit…

A:

Q: Find all the primitive roots modulo 125.

A:

Q: 12. Prove that there are infinitely many prime numbers in the arith- metic progression 1,4,7, 10,…

A:

Q: Every two different prime numbers are relatively prime. False True

A: In this question we need to check the given statement is true or false

Q: Prove or disprove: for every natural number n,2+5+8+...+(3n-1)=n(3n+1)/2

A:

Q: Prove: The nth Fibonacci number Fn is even if and only if 3 | n.

A: The nth Fibonacci number Fn is even if and only if 3|n.

Q: Find the sum of the positive even integers up to and including 50.

A:

Q: Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for…

A:

Q: There exist two prime numbers and such that their product gives 1261

A: We have to find two prime numbers whose product gives the required number

Q: Every integer nz2 either is a prime or can be expressed of a product of primes. This statement is…

A: From Fundamental theorem of Arithmetic says that Every positive integer n>1 can be expressed as…

Q: Prove by Induction that for every natural number n, 3n-1 is a multiple of 2? Attach File

A: Introduction: Divisibility is a very important property in the set of integers. A number can either…

Q: Prove that 3 | 22n+1 +1 for all natural numbers n.

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

Q: Find the smallest positive integer for which x mod 3=2 and x mod 4=3   What is the next smallest…

A:

Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

A:

Q: Find the smallest integer i for which the statement (1 + i2 )<2' is true. Then, prove the statement…

A:

Q: Prove that (5n)!÷40^n×n! is natural number for every natural number n​

A: To Prove that (5n)!÷40^n×n! is natural number for every natural number n​

Q: Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

A:

Q: Assume that all four-digit integers (numbers from and including 1000 to and including 9999) are…

A:

Q: Determine the number of ways a computer can randomly generate one or more such integers from 1…

A: There are 5 prime numbers between 1 to 12 . Which are 2 ,3 , 5, 7 , 11.

Q: Prove by Mathematical Induction that for any natural number n the number 52n+1 – 22n+1 is divisible…

A:

Q: Show that if n is an odd integer then []

A:

Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: Consider all numbers between 20000 and 70000. Efficiently determine: A) How many of these numbers…

A: Part A - Let’s represent the number as ABCDE. You have to considere the special cases first, since…

Q: 15. Prove that for each natural number n, s+ (4n-3)(4n+1) 4n+1 ...

A: Given The given expression is 11·5+15·9+·  ·  ·+14n-34n+1=n4n+1

Q: Prove the rule of exponents              (ab)n= anbn by using principle of mathematical induction…

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: Prove that every natural number n can be written in exactly one of the following forms: • 3k for…

A:

Q: (5n)! Prove that 40"-n! is natural number for every natural number n.

A:

Q: How many numbers from 1-999 have: 1) all three digits the same? (assume that "1" is represented as…

A:

Q: Show that the principle of mathematical induction implies that I is the leact p natural number.

A: Solution of the problem as follows

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

A:

Q: Identify the real fourth root(s) of 256. O -4 O -2 O -1 O 2 O 4 O 2 and –2 O 3 and -3 O 4 and -4 no…

A:

Q: (5n)! 40 n! is natural number for every natural number n. Prove that

A:

Q: Question 5 (5n)! is natural number for every natural number n. 40" n! Prove that

A:

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

A:

Q: Prove that the sum of the first n natural numbers is given by this formula: n(n + 1) 1+2 +3 +. ..…

A: As per our guidelines, we are supposed to answer only one question. Kindly repost other questions as…

Q: show that for every positive integer n there is a prime whose decimal expansion contains n…

A: so we have to prove , for every positive integer n there is a prime whose decimal expansion contains…

Q: Prove that each statement is true for all natural numbers. 1) 3+7+11+...(4n-1)=2n^2+n

A: Since you have posted multiple questions i can do first question as per our company guidelines…

Q: A man finished a job in 5 days. On the first day, he finished 1/m of the job. On the second day, he…

A: GIVEN THAT JOB FINISHED = 5 DAYS

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.