If a is an algebraic integer satisfying a³ + a + 1 = 0 and ß is an algebraic integer satisfying B² + B-3 = 0, prove that both a + B and aß are algebraic integers. %3D

Q: 13. If a is an algebraic integer satisfying a + a + 1 = 0 and ß is an algebraic integer satisfying…

A: Given: α is an algebraic integer satisfying α3+α+1=0 and ß is an algebraic integer satisfying…

Q: If x is any positive integer , then the sum of 6x and 11x is always divisible by what?

A: x is a positive integer. Find sum of 6x and 11x is divisible by what?

Q: If a, b. and c are integers such that a|b and c|b. then ac|b.

A: False

Q: If k is an integer: exp[(i)(pi)(2k+1)] = ?

A: Given that: exp[(i)(pi)(2k+1)]

Q: show that if x is an integer then x2+x-41= 0 produce prime numbers

A: Here show that if x is an integer then  produce prime numbers.

Q: Let a be an odd integer. Show that (i) 8 | a2 − 1; (ii) 16 | a4 + 15.

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Q: Determine k so that; is singular.

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Q: If 5|(2a + 1) for some integer a, then 25|(14a² + 19a – 19).

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Q: For positive integers a and b, which of the following has the same value as gcd(a, b)? O a-b O a+b O…

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Q: Find n if: a) P(n, n) = 5040. b) 66= C(n, 2). c) P(n+1, 5)= 5P(n,1) Note: n is a…

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Q: Every natural number n > 2 cam be written n = P1P2 Pr as the ... product of distinct prime numbers…

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Q: The product of two integers of the form 6k+5 is of the form 6k+1 Select one: True O False

A: 1. The word integer originated from the Latin word ''Integer'' which means whole. It is a special…

Q: If is an odd integer, show that n* + 4n2 + 11 is of the form 16k.

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

Q: , Let a be a positive integer, where the positive difference of every two digits of a in base 10 are…

A: Here, For 1113,     | 1-1 | =0 , | 1-3 | = |-2| = 2 For 2000,     |2-0|= 2, |0-0|= 0 For 2022,…

Q: Show by contraposition that for positive real numbers a and b, if ab - a - b ≥0 then a and b are…

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Q: Let n be an odd integer, a > 0, and b < 0. Prove that x" + ax + b= 0 has a unique root. Hint: If…

A: Let n be an odd integer, a &gt; 0, and b &lt; 0. Prove that xn + ax + b = 0 has a unique root

Q: For any integer n ≥ 1, C(6n, 3n) C(6n, 3n-1). O True O False

A:

Q: Prove that the value of an expression (a - 1)(a - 3)(a - 4)(a - 6) + 10 is positive for every a…

A: To prove that   the value of an expression (a - 1)(a - 3)(a - 4)(a - 6) + 10 is positive for every a…

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: Every positive integer is the sum of at most two squares and a cube of nonnegative integers. O True…

A: consider 0  0=02+02+03 now  1 can be written as  1=02+02+13 now consider 2 2=12+02+13 now consider 3…

Q: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: 5. Let a be a real number, c an integer, c > 0. Prove that

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Q: Find all possible values of P in x^2+10x+P=(x-A)(x-B) if P, A and B are all whole numbers

A: To find all possible values of P in x2+10x+P=(x-A)(x-B) if P,A and B are all whole numbers.

Q: If AB = BA and p is a nonnegative integer, show that (AB)p = A p B p .

A: Given p is non-negative integer AB=BA

Q: For an arbitrary integer a, verify the following: (a) 2|a(a + 1), and 3| a(a + 1)(a + 2).

A: We have to verify, 2|a(a+1)  and 3|a(a+1)(a+2) Here we use division algorithm.

Q: let a and b be integers. Show: if a * b is odd, then a is odd and b is odd

A: A number is even, if it is divisible by 2. A number is odd, if it is not divisible by 2. The odd…

Q: a) Evaluate each. 10! 2. 3! 7! 5! 3. Р(5, 3) 4. Р(6, 6) 1. 4! b) true or false, where n is an…

A: It is known that: n!=n·(n-1)·(n-2)·⋯·2·1Pn, r=n!n-r!

Q: If a is any integer, is a(a + 1) even or odd?

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Q: Let a be a positive real number and let n be an integer greater than 1.Prove or disprove that…

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Q: For an arbitrary integer, say A, verify that 2|A(A+1). Similarly, verify that 3|A(A+1)(A+2).

A: To verify:A(A+1) is divisible by 2.

Q: If a and b are distinct integers prove that a-b is a factor of an -b" whenever n is positive integer

A: if a and b are distinct integers prove that a-b is a factor of a^n - b^n whenever n is positive…

Q: For positive integers a and b, which of the following has the same value as gcd(a, b)? gcd(a+b,a-b)…

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Q: a. If b is any nonzero real number, prove that |1/b| = 1/| b | . b. Prove that |a /b | = |a|/| b|…

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Q: If k is an integer: exp[(i)(2pi)(k)] =

A: Solution) ei 2ℼ k As we know eia = cos a + i sin a a = 2ℼk  ei 2ℼ k= cos 2ℼk  + i sin 2ℼk  K =…

Q: B={xx is an integer and-1<x<1}

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Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

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Q: prove that if a and b are both odd integers, then ab+1 is an even integer

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Q: If a²is an even integer, then a is an even integer. If d = ax + by, then d = gcd(a, b). If d|a + b,…

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Q: Let p = p1 ... Ps be an integer with digits p1,..., Ps. Show that 0.p1 . . · Ps %3D 10° – 1

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Q: If a, b, and c are integers such that b is a multiple of a and e is a multiple of b², then c is a…

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Q: Let, a1 = 3, a2 = 4 and for n > 3, an = 2an-1+an-2 +n², express an in terms of n.

A: a1=3, a2=4 n≥3; an=2an-1+an-2+n2 First from the given, the given values shows that the terms are in…

Q: How many nonnegative integers a, b, c.2 and d are there that satisfy the equation 1 + + a+b 1 a…

A: Given equation has 4 variables a,b,c,d Sum=1 We can find pairs by the following ways

Q: For all positive integers a, b, where a < b, if a {b, then gcd(a, b) = 1. O True False

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Q: Show that for any integer n ≥ 2, C(2n,n) > C(2n,n-1)

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Q: Prove that for integers a and b, (a -3)b2 is even if and only if a is odd or b is even.

A: We need to prove that for integers a and b, (a -3)b^2 is even if and only if a is odd or b is even.

Q: Let A = {x | x is a integer and x^2 < 16}, identify each of the following as true or false:

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Q: For all integers a, b, and c, a | bc if and only if a | b and a | c. ect one: True False

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Q: Let i be a positive integer and A, = [-i+1,1[. Then N2, A,- None of these O 10.11 [0,1[ 10.10

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Q: If n is an odd integer, show that n4 + 4n2 + 11 is of the form 16k.

A: Solution: The objective is to show that given equation is of the form 16k

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.