21. Find two different factorizations of the number 21 into primes of the form a + b v-2, where a and b are integers.

Q: 20. (a) Show that the sum of two odd prime numbers is never prime. (b) Is (a) true if the word odd…

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: 2. Find six rational numbers between 3 and 4

A: We can solve the given problem using concept of rational numbers which is, Therefore d=(y-x)/(n+1)…

Q: 19. What is the value of (pk) when p is prime and k is a positive integer?

A: We know that the value of the Euler ϕ-function at the positive integer n is defined to be the number…

Q: 2. If b and c are relatively prime and a / bc then a / b or a /c.

A: First Method: Since b and c are prime: gcd (b,c) =1 Then: gcd (ab, ac) is equal to "a" Then: Let for…

Q: What can you conclude about the numbers m and n if X " = 7 and r is an even number and m and n are…

A: The number that is divisible by 2 is called the even number and the number that is not divisible by…

Q: If n has a prime factorization n = pi*p2² ... p" then the sum Eajn d³P(d) is equal to: a1. az

A: Euler's phi function is widely used in number theory and algebra. This function represents the…

Q: Justify that every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q.

A: Let p be a prime number other than 2 and 3. If p is divided by 6, then by the quotient-remainder…

Q: ..y yun.. 40mA B: Give the prime factorization of the number 34417

A: 34417= 127×271 The only divisors of 127 are 1 and 127 itself this implies that 127 is a prime…

Q: If a, b are two distinct prime number than a highest common factor of a, b is: (2 ab

A: HCF of two numbers is found out by taking common prime factors from both the numbers the…

Q: Suppose N = Fox F₁x F2, where the three factors are pairwise prime; hence, each factor and the…

A:

Q: 2. Find the missing numbers in the magic square below. 2 13 10 11 6. 12 15 1 4-

A: Consider the provided table,

Q: b. The location of 0.47 is between which two hash marks on the number line? and Which of these two…

A: The location of 0.47 is between two hash marks on the number lies 0.4 and 0.5. The 0.47 closet to…

Q: The number of factors of prime numberss are (None of the mentioned Depends on the prime number (2 (3

A: Prime number is only divisible by 1 and that number itself

Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

A:

Q: Characterize the primes p for which 63 is a quadratic residue.

A: Quadractic residue: Let p  be any odd prime number and gcd(a,p)=1. Then a is said to be a quadratic…

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

A: Prime and divisibility

Q: For which prime p,  p^2+2 is also prime?

A: Put p=3 in p2+2 and obtain that, 32+2=9+2=11. Clearly, 11 is a prime number. Thus, p=3 satisfies…

Q: 10. If the product of N consecutive posi- tive integers is divided by N, the remainder is always (A)…

A:

Q: Show that if 5 divides n2 with remainder 4, 5 divides 3n3 with remainder 2.

A: Given that 5 divides n2 5 n25 n×n5 n5 n3 As it gives the remainder 4 on dividing by 5. Therefore it…

Q: Show by mathematical l Induction y 4n-2 3. +17 can be divided by +22 possitive integers 16 far all

A: To prove the given statement by using mathematical induction.

Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: If n is any integer then n and n5 will have? (mcq) A. Both are divisible by 5 B. 10s place is…

A:

Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: Show that we can easily factor n when we know that n is the product of two primes, p and q, and we…

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: Suppose that m, n are two integers such that m 2n2 = 1. Does it follow that m and n are relatively…

A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

Q: 3. The GCF of any two distinct prime numbers is 1. (Distinct they are not the same number) means

A:

Q: 37! D- Given that N= (191)(15) is an integer: i- Does the prime 11 divide N? What about the prime…

A: N = 37!19! 15!37! consist numbers that are divided by 11→11, 22 , 33→319! consist numbers that are…

Q: 2 (a) Work out 14 x 4 – 24 ÷ 2. (b) Find a number between 80 and 90 that is a multiple of both 3 and…

A: We have to solve two subparts in the given problem.                        (a) We need to work out…

Q: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

A: Given: p≥q≥5 and p and q are both primes

Q: Suppose that m, n are two integers such that m3 – 2n2 = 1. Does it follow that m and n are…

A:

Q: a.For the 2019 Comrades marathon, 17268 men and 4357 women entered, 15835 men and 3982 women started…

A: As per bartleby guidelines for more than one question, only 1st one is to be answered. Please upload…

Q: 2. What is the prime factorization of 15x y? O 3.5 2y O 3.5 x y Y O 15 - x * y Y 3-5 22 y y

A:

Q: If m is a positive integer, does 6 divide (2m)(m+2)(3m+3)? True False

A: Here, the given number is: (2m)(m+2)(3m+3)=2(m(m+2))3(m+1)=6m(m+1)(m+2) because multiplication is…

Q: Find all prime numbers pand of for which 2p²q + 45 pp ² = y² 2 де

A: Given:

Q: (6) Find the sum of the integers for 7 to 508 Tmelusive,

A: first term (a0) = 7last term( an) = 508,     d = 1 ( natural no.)first we have to find total no of…

Q: (12) What values can be assigned to r so that {4n+r} will include all odd primes?

A:

Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 +q2 + r2.

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: Show that the sum of twin primes p and p + 2 is divisible by 12, provided that p> 3.

A: Let p, p + 2 be twin primes. Then p *(p+1) + 1 = p^2 + 2p + 1 = (p+1)^2 ,  a perfect square.

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: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

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Q: 8. Show that no prime number of the form 4k +3 can divide a number of the form n2 + 1.

A:

Q: Suppose that m,n are two integers such that m3−2n2=1. Does it follow that m and n are relatively…

A:

Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 + q2 + r2.

A: We have to prove that 3 divides p2+q2+r2, where p,q and r primes other than 3.

Q: 4. Suppose that a and b are integers that divide the integer c. If a and b are relatively prime,…

A: Suppose that a and b are integers that divide the integer 'i'If a and b are relatively prime. Show…

Q: 4. Find them if: (a) lcm of two consecutive positive integers is 812; (b) lcm of twin primes is 899.

A:

Q: 4. The area of a parallelogram is 198 square units. If the base and the height of the parallelogram…

A:

Q: 1.What is the smallest number that can be exactly divided by 2, 3, 4, 5  and 8 ? 2.What are the…

A: To answer the given questions.

Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.