A-(kikis a positive integer and 22

Q: if 3 does not divide n then 3 divides 2n^2+1

A: Let us Assumue 3 divides (2n2+1) n2+1=3λ Now  2n2+1=n2+1+1⇒(2n2+1)+(-λ)3=1⇒ gcd(2n2+1,3)=1…

Q: 2) x" = (Power RUle, n any nonzero integer)

A:

Q: З and an - - 3ап-1 + an-2 for integers n>2. Evaluate and d4: Let a, = 2, a2

A:

Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

Q: The principle of mathematical_______ states that a statement involving positive integers is true for…

A: Consider the given fill in the blank:   The principle of mathematical induction states that a…

Q: 4 13 a) What is the largest natural number n such that- 11 7 b) Using the ideas from part a, find…

A:

Q: rix with even mumber of permuta Q4. If all elements of A and A are integer numbers, what are the…

A: Let A be any n×n Matrix with integer entries

Q: 4. For the incquality 3n+1> n° determine the set of natural numbers for which

A: Introduction: For all-natural numbers, the principle of mathematical induction is a concept that…

Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: Suppose x and y are integers. Show that If x y, then x | 3y - y + 5y. 1) What is your assumption? 2)…

A:

Q: Suppose ne N. Then the following equation is true: () - ) -)-) - + (-1 = 0. + 3 How can this…

A: Let's find.

Q: Choose what method of proof to use always provide solution.   a) Prove that if the sum of any two…

A:

Q: -, 1+2+2 '+.+ 2 -1 for all natural numbers n.

A: We will use the principle of mathematical induction to prove the result.

Q: p + 2 is a prime number, prove that p2+3 is divisible by 4. (2.5

A: Prime and divisibility

Q: 3. Let n be an integer greater than 2. Explain why n! +2, n! +3, n! +4, n! + (n – 1), n! +n are all…

A: We have to explain that n!+2, n!+3,…,n!+n-1,n!+n are all composite numbers.

Q: Find an integer such that its square is 81 less than 18 times the integer

A:

Q: Modal marks for a group of 94 students are 54. Ten students got marks between 0-20, thirty students…

A:

Q: Let n be a number with 3 digits. (i.e 100sns 999) Show that if the sum of digits of n is divisible…

A:

Q: Find all the odd positive integers n which satisfy ó(n) = 20. %3D

A:

Q: 10. Determine three positive numbers whose sum is 48 and such that their product is as large as…

A: find the three positive number.

Q: Let A2A2 be the set of all multiples of 2 except for 2.2. Let A3A3 be the set of all multiples of 3…

A: Let A and B be two sets. Then the union of two sets will be the set that will contain all the…

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

A:

Q: 16. What is the greatest integer N such that (10 – N) + (16 - N)> 5?

A:

Q: 5.( Use the methods of this class to find two numbers, x and y, whose sum is 10 and with the…

A: A series is the set of numbers may be related by some rule, which is not a necessity. Series may be…

Q: Give a combinatorial proof (an algebraic proof will not be accepted ()()-(")- (;) 2n n 2 . -2. 3 3

A:

Q: 3. How many zeros at the end does the number 2022! have? (n! is a product of all positive integers…

A: Given,Number = 2022!To find: Number of zeros at the end (trailing zeros) of the given number  As per…

Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: Suppose that m and n are real numbers with m + 5n = 10. Find the least possible value of the…

A:

Q: 2. Show that if a]b then a'|b² for all integers a and b (Symbols V, 3, A, V, V,A,→,-)

A:

Q: s) Express the infinite contimod fraction [2, 1,5 quadratic irrational- as a ity.

A: We need to express the infinite continued fraction 2, 1, 5¯ as a quadratic irrationality. The…

Q: If a and b are integers with gcd(a, b) =d Then gea (-) -- gcd

A: False

Q: Let n = integer; 7 divides n if and only if 7 dividesn^2 (The result(s) ought to be integer(s))

A: Consider an integer n. We have to prove that 7 divides n if and only if 7 divides n2. Suppose that 7…

Q: Prove (n + 6)² – (n + 2)² is always a multiple of 8. (Use direct proof.)

A:

Q: How do I prove the relative prime relationship only given that for some integer value exists for a…

A:

Q: A positive integer is 4 less than another. If 3 times the reciprocal of the smaller integer is…

A:

Q: k is a prime number that has a ones digit of 7 and 0 <k < 100, how many values of k are possible?

A:

Q: 2. Suppose that your scores on the first four tests in a course are 75, 82, 60, and 91. There is one…

A: Given score of four tests is 75,82,60,91

Q: a) Prove that the sum of the squares of two odd integers is always even.

A:

Q: Please help me to write this as a proof.  Let n be an integer. If n is the sum of three consecutive…

A:

Q: f repetition is not allowed and must not leading with a zero, how many different four digit numbers…

A: we have to find the no of many different four digit numbers.

Q: Prove that 2^? +3^? is a multiple of 5 for all odd ?∈N.

A: We will solve this problem using principle of mathematical induction.

Q: 9 - K) = 1, Which is impoSsible. Which proof method is used in above proof. Proof by counterexample…

A: Given that: If x and y are two even integers, then their sum is even.

Q: 4. ( it as linear combination of both numbers. Find the greatest common divisor of the numbers 1520…

A:

Q: 3. There are seven students who always compare their CS test grades. They discovered that either the…

A: It is given that there are seven students who always compare their CS test grades. They discovered…

Q: yields a true statement when n is replaced by any positive integer.

A: The above statement is true. Step-2 i explained in detailed explanation

Q: Number theory

A: Assume a=x2

Q: Use the Parity Theorem to prove: For all integers n, n2+3n is even

A: Given: n is integer. To show: n2+3nis even.

Q: [1] Proof by case: a) Any two consecutive integers have opposite parity. b) Prove: For any integer…

A:

Q: Abby notices that for the first twenty perfect squares, each square is either a multiple of 5, one…

A: If n is a natural number, then n2 can be written as 5k- 1,5k, or 5k +1, where k is a whole number.…

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!