1.1 What are each of the following? (a) |{0, 1, 2, 3, 4, 5, 6}| (b) (c) 111 (d) The set of divisors of 100. (e) The set of prime divisors of 100.

Q: 3. How many 4-digit positive integers are there that are even or contain no O's?

A:

Q: The number of factors of prime numbers are None of the mentioned O 30 Depends on the prime number

A: No. Of factor of prime no. Means how many divisor consist of a prime no.

Q: Let f(sub-n) denote the nth Fibonacci number. Show that f(sub-n) is even if and only if n is…

A:

Q: 1. Compute the value of (), for n > 3.

A: Formula: nr=n!r!n-r!

Q: 1. Show that subtraction is not associative for the set of integers.

A: Definition :Let "." be a binary operation and A is a setThen "." is said to be associative for A if…

Q: 1. Prove that 7" -1 is divisible by 6 for all n > 1.

A: As per Bartleby Guidelines we can solve only one question at a time. Hope you understand

Q: 8. How many positive integers less than 1000 a) have exactly three decimal digits? b) have an odd…

A: According to our guideliness we can answer only the first 3 subparts. Positive integers less than…

Q: . Show that n! > 3ª for n> 7.

A:

Q: Find how many positive integers with exactly four decimal digits, that is, positive integers between…

A:

Q: 3. Let n be an integer, show that if n is even, then n is also even using indirect proof.

A: Given : Let n be an integer.   To show that if n2 is even, then n is also even.

Q: 9. This question is a little more challenging. How many integers in the interval [1, 99 999] (i.e.,…

A: Given data:   The given sum of all digits is k=10. The number of digits that can be used is n=5.…

Q: Q1: what's the relation between prime and semiprime ideals.

A: As per the guidelines we are supposed yo solve the first question.kindly post another questions…

Q: How many integers greater than 5400 and less than 10000 have both of the following properties: -…

A: To count the number of positive integers satisfying all the required properties

Q: Number Theory

A: Given that n is a positive integer. The objective is to prove that

Q: - (Test for Divisibility by 11) Let n be an integer with decimal repre- sentation a,a, if a, – a, +…

A:

Q: 3. Consider the word PERMUTE and all the ways you can arrange its letter using each letter only once…

A:

Q: 5. How many positive integers between 50 and 100 a) are divisible by 7? Which integers are these? b)…

A:

Q: 1. Consider the following rules on the set of natural numbers, N, 1) write down 5 ordered pairs of…

A: The 5 order pair of the natural number that is satisfy the relation “less than or equal to” as…

Q: How many positive integers between 50 and 100 a) are divisible by 7? Which integers are these? b)…

A:

Q: 3. Prove that 2" > n For all nuatural numbers

A: Solution

Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

A:

Q: イの Prove by mathematical induction that 3" + 7 is divisible by 8 for all integers n > 0

A:

Q: 24. How many of the following equalities are correct? 3.1459 ÷ 10-5=14590,0.0075 × 10³ = 75, 4567 x…

A:

Q: 44. Which of these collections of subsets are partitions of the set of integers? a) the set of even…

A:

Q: 5. Give a recursive definition of the set of positive integers that are multiples of 5.

A: Solution

Q: 9. Find the sum of all'integers between 90 and 190 if each integer Is divisible by 17 (Hint. The…

A:

Q: 1)How many integers between 1000 and 2000 are divisible by at least one of the integers 3, 5 and 7?

A: Introduction: In mathematics, integers are the sum of positive and negative numbers. Integers are…

Q: 27. Determine the number of n-digit numbers with all digits odd, such that 1 and 3 each occur a…

A:

Q: 3. If а, 1000". Find N E I, the set of positive integers, such that an+1 N). n!

A: Given an=1000nn!. We have to find the N∈I such that an+1<an, n>N. For n=1 a1=100011! For n=2…

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

A:

Q: How many positive odd integers less than 10,000 can be written using the digits 3, 4, 6, 8, and 0?

A:

Q: 12 belongs to the set of natural numbers

A: The natural numbers (i.e. counting numbers) are numbers which are used for counting and ordering.…

Q: Show that n! > 3ª for n > 7.

A: The given inequality is, n!>3n We need to prove that, the inequality is true for n≥7 Firstly,…

Q: 15 Enter your answer in each of th

A: Here we use properties of a right angled triangle to find height.

Q: Show that n! > 3 for n ≥ 7

A: We have to prove this assertion that, n! > 3  for n≥7 We will prove this statement by the method…

Q: 11. Establish the following relation between Fibonacci and Lucas numbers: L, = un + 2un-1 for all n…

A: Remark: We know that both the Fibonacci numbers and  Lucas numbers are obtained as the sum of its…

Q: 5) Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4.…

A:

Q: Prove the following:For every natural number n≥1, 1+4+7+⋯+(3n−2)=n(3n−1)2.

A: To prove the identity involving the sum to n terms of the given series

Q: 21. Prove that if n and k are natural numbers such that k < n, then (*:) - (-) - () n k k

A:

Q: What is the largest n such that 16 and 25 are in the same remainder class modulo n? Write out the…

A:

Q: Let P be the set of integers that are multiples of 3 between 1 and 10 and Q be the set of positive…

A:

Q: How many 5-tuples (x1, x2, x3, x4, X5) of positive integers from 1 to n are there? How many 5-tuples…

A: Since you have asked multiple questions, we will solve the Question Q1 (a) and (b) and (c) for you.…

Q: 24. Suppose that I choose six of the first ten positive integers. Prove that I must have chosen two…

A: Rewrite the first ten numbers as follows.

Q: 10. A small population of 27 birds on an island are marked so that each individual can be…

A: Combination

Q: (1). Prove that adding an irrational number with a rational number is an irrational number, and show…

A: Note:-Our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: ove that n’ +5n is divisible by 6 for all n E N.

A:

Q: How many different orders are possible in Z*92639 (Hint: 91639 is a prime; 91638=2 . 27 . 1697).…

A: as 92639 is a prime number so Z*92639 is cyclic group of order 92638.

Q: Prove or disprove the following: n+1 For any odd integer n, H= ()) 2

A: Let  n=2k+1         -  (1) Where k is an integer. Then n is any odd integer. Now, LHS= n24        -…

Q: 7. Consider the following six numbers: 4 7 -1.23 , v2 , 3 , n , 7'11 (a)- Which of the above…

A: We have to consider the  given numbers

Q: 5. Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  100≤n≤999 such that the last three digits of n2 is same as n.

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!