Denote by an the nth positive integer that is not a perfect square. For example, we have n-perfect-square integers {2,3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, ...}, and then -1 = 2, %3D a2 = az = 5, a4 = 6, a5 = 7, 8, a7 = 10, a8 = 11,... a6 = an = n + {/n}

Q: Prove that Va - Vb is irrational number if 'a' and 'b’ are prime number?

A: Given To prove that a-b is irrational number if a and b are prime number.

Q: are there any even numbers greater than 2 which are prime? why or why not?

A: We will find the solution in the following step

Q: True or false: If n is a positive integer with d(n) = 6, then the only possible prime power…

A: n is a positive integer with dn=6.

Q: If a,b,c,d are in distinct real number in A.P. Such that 2(a-b)+c(b-c) +(c-a) =2(a-d)+(b-d) +(c-d)'…

A:

Q: Show that if a positive integer is odd, then it's square is odd.

A: To show that if a positive integer is odd then its square is odd

Q: determine the value of c(n,5) /p(n,5) where n can be any integer thats 5 or greater

A: Recall the following. Permutations Number of permutations of n things taken r at a time: pn,r=n!n-r!…

Q: How many 6-digit numbers have three even and three odd digits? (The first digit cannot be zero.)

A: It is given that a 6-digit number is to be formed which will have three even and three odd digits.…

Q: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

Q: o prove that 99 is not the square of a positive integer, we just need to consider three cases for an…

A: The objective is to determine if the given statement is true or false. The statement is  to prove…

Q: Add the following integers: 1. -90 + (-13) = 2. 8 + 4 = 3. -17 + 122 = 4. -312 + 211 = 5. 19 + (-18)…

A:

Q: bers {1, 3, 6, …}

A:

Q: 1. Prove that the sum of an integer and its square is even

A:

Q: prove that the sum of three odd integers is odd using indirect method. show your solution and…

A:

Q: How many integer solutions to x1 + x2 + x3 + x4 x₁ ≥ 4, x2 ≥ 2, x3 ≥ 5 and x4 ≥ 3? > = 31 are there…

A: Find change all inequality to grater then equal to 0. Then use the theorem

Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

A:

Q: 5) Let n be a positive integer such that 7 | (8-1). Prove that 7 | (8=*1 - 1). (Hint: Rewrite 8*1…

A: Given: Let n be a positive integer such that 78n-1. To prove: 78n+1-1

Q: Prove that 67n+ 17 cannot be a perfect square for any three consecutive integers n.

A: We will use quadratic residue and proof by contradiction method.

Q: Prove that (a) For all integers m and n, n + m and n - m are either both odd or both even.

A: Since you have posted multiple questions, according to our guidelines we can solve only first…

Q: If a and b are integers with gcd(a, b) = d Then a gcd %3D -1

A: False.

Q: What is the largest number x such that x<200 and x has exactly 4 factors?

A: Factors:    Number that are multiplied together to get a product                              Factor…

Q: Prove the following using a direct proof.  If a,b, and c are integers such that b is a multiple of…

A:

Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's Theorem: If p is a prime and a is a positive integer and not divisible by the prime p, then…

Q: Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5

A:

Q: hat is the last digit in the product of (3¹)(3²)(3³)(34)...(34⁰⁰)?

A:

Q: Show that, for an integer n, if n^3 is odd, then n must be dd.

A:

Q: pls help asap!!!

A: According to guidelines we solve first three subparts when given multiple subparts.

Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's states that if p is a prime and a is a positive integer not divisible by p, than ap-1≡1 mod…

Q: 2. Find 14 --3-6 -8 7) 4

A: To find the product matrices.

Q: How many consecutive zeroes are there at the end of 100! (100 factorial). For example, 12! -…

A: How many consecutive zeroes are there at the end of 100! We know that…

Q: Prove that if x is an even integer, thenx^2−6x+ 3is odd

A: Any integer of the form 2m+1 is odd.

Q: prove the difference of the squares of two z integers is always divisible by 16 (i.e. z1^2 - z2^2 =…

A: We prove that (z_1)^2-(z_2)^2=16k for some integer k, then we say that divisible by 16.

Q: Let ß = (1,3,5,7,9,8,6)(2,4,10). What is the smallest positive integern for which ßn = ß-5?

A:

Q: Find the 66th term of the triangular numbers {1, 3, 6, …}.

A:

Q: We select 38 even positive integers, all less than 1000. Prove that there will be two of them whose…

A: Given statement is Here we can select 38 even positive integers all are less than 1000. let…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.