3) Let II" , m i=1 where p1 < p2 < ... < pi are prime numbers and a; > 0 integers. Show that d eN is a divisor of m if and only d = II p§ª with c; an integer, such i=1 that 0 < c; < a; for all 1 < i 0 are integers then %3D i=1 min(a;,b¿) gcd(m, п) i=1

Q: Evaluate f1 dy Vy-1 O a. 2ỹ+ 2ln(ỹ – 1) +C O b. 2j – 2ln(/ – 1) + C O c. None\ of\ the\ following O…

A:

Q: 6. Let a be a number written (in base 10) as ao 10⁰+ a₁ 10¹ + a₂. 10²+...+an. 10" where 0 ≤a < 10.…

A:

Q: Find the intersection between the set of all integers bigger than 0 but smaller than 10 and the set…

A:

Q: For all integers a, b, and c, if a/b and a|c, then a|(5b – 7c). -

A:

Q: Which of the following is an example of an infinite set which is bounded and countable? * A.) The…

A:

Q: Find the complete set of all positive integer numbers such that each of them is 13 times greater…

A:

Q: 2. U={x|x is an integer, 2<x<10}. A and B are subsets of U such that A= {multiples of 3}, B={factors…

A: Using set theory we have to answer following.

Q: * 63. Let a¡, az» ·… , A, be positive real numbers. The arith- metic mean of these numbers is…

A: The arithmetic mean AM=a1+a2+...+annThe geometric mean GM=(a1,a2...,am)1n We need to show A≥G.

Q: fa,,a,,a,,...-,a, be an AP of non - zero terms, prove that ---- 1 +. 1 +...+ n-1 aa, a,a, a a, a,a

A:

Q: Prove that Vn e N (the set of natural numbers), ÷+ ɛ N. 3 +

A: We have to use proper manipulation.

Q: Show that there exist no two positive integers nandr(r<n)such that "C, "C,+1 "C+2 and "C,.3 are in…

A:

Q: For each integer n ≥ 1, let pn be the number of legal strings with n letters that also read the same…

A: The correct expression can be found by understanding the pattern of palindromic strings. Palindromic…

Q: Let Find formulas for the entries of M", where is a positive integer. M" O M -2*5**n 10 5 -5*n

A:

Q: Z+ is the set of positive integers. Let A = {1,2,3), B = {n ∈ Z+ | n is even}, C = {n ∈ Z+ | n is…

A: concept Used: C∩D is the set  containing all elements of C that also in the set D C∪D is the set…

Q: If fN is the set of the first 100 positive integers, A is the set of odd numbers between 50 and 90,…

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

Q: Let S be the set of all real numbers except -1. Define * on S by a*b = a + b + a b. The solution of…

A:

Q: 1. SCP or SSxy" 2. SS,=. 3. SS,=, 4. Calculate the Pearson product moment correlation between X and…

A: Since you have posted a question with multiple subparts, we will solve first 3 subparts for you. To…

Q: Prove that, all integers x and y, x^2-4y≠2  1a. Prove that all numbers of the form 1007, 10017,…

A:

Q: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

A:

Q: 5. Let r,ne N. Show that the product (n+1)(n+2)...(n+r) of r consecutive positive integers is…

A:

Q: How do you prove that ∀x∈Z,n,m∈N(xn)m = xmn ? (where Z is set of integers and N is set of natural…

A: Given x∈ℤ, n,m ∈ℕ xnm=x·x·x·x·x·⋯·x⏟m                         n times…

Q: b. How many elements are in the set {a, {a, b}, {a} }? C. For each positive integer x, let A1` =…

A: Use the definition of set and its elements to solve this problem as in next step.

Q: Let S, be the set of real numbers such that their decimal ex- pansion contains 3 at l-th place after…

A:

Q: 10. Let m and n are two positive integers with m > n. Let a be a positive integer. Prove that a2" +1…

A:

Q: 4c Let m and n be integers. If m +n is even, then either ma even or m and n are both odd. 10 = 2F+1…

A: Consider the following statement: For all integers m and n, if m+n is even then m and n are both…

Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

A:

Q: Let m%3= i=1 where pi 0 are integers then i=1 i=1 gcd(m, n) = I min(ai,bi). %3D i=1

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

Q: Every none-empty subset of integer numbers contains a least element.

A:

Q: Prove that for every real number x, 3x - 6 ≤ x iff |x-1| ≤

A: The objective is to prove that , where  is a real number.The inequality  is equivalent to .

Q: Prove that for every positive integer n, there exist i and j with 1≤ i < j≤n +1 such that n is a…

A: Answer

Q: 4. Let R+ be the set of positive real numbers and N+ be the set of positive natural numbers. Prove…

A: The objective of this question is to prove that for all positive real numbers a and b, and for all…

Q: Prove that 1.3 + 5.7 + 9.11 + =

A:

Q: 7. Without using any theorems from the book, prove the following statement: Let n be an integer. If…

A:

Q: 9. Ifn is a squarc-frce integer, prove that r(n) = 2" , where r is the number of prime divisors of…

A:

Q: Let {\mathcal S}S be the set of all (nonnegative) 4-digit integers. How many integers does {…

A: Here given S is set of all (nonnegative) 4-digit integers We have to calculate,  How many 4 digits…

Q: Prove that a selection of 7 numbers from the set {1, 2, . . . , 11} contains numbers m and n such…

A:

Q: 12. A real number x is called algebraic if there is a positive integer n and integers aŋ, a1, 7 0.…

A:

Q: 13. Pier Fermat (1601 – 1665) conjectured that any number F, of the form 2²" + 1 where n is…

A: Fermat number is Fn=22n+1