1) Coirect Proo8) Fove that the produck of two whsecutive integers ís divisible by 2. 2. (indirect proot) Prove that if a and o ae buth positive, ard az <62 t

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: 2show that O (ir Yational numbers iihot hegigitale

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Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

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Q: theorem Every Ving Can be imbeobling în aving with

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

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Q: Assume an unlimited supply of coins worth 5 cents and 9 cents. Prove that some

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Q: Prove that 1+1=2

A: To prove: 1 + 1 = 2

Q: Prove that well prooluct of Sum as as amy two conjugate complex numbers is real number-

A: Let Z1 and Z2 be two complex numbers such that  Z1 = a1 +ib1 and Z2 = a2 + ib2  Where a1, a2, b1, b2…

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: Prove (kindly provide reason/s) that for any real numbers m and n, (-m ) + ( - n ) = - ( m + n)

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Q: (b) Prove that 27 (2⁹⁹ +1).

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Q: Prove the following using either direct proof, indirect proof, or proof by contradiction. 1. If n…

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Q: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

Q: Let a > 0 be a real number and I = [" LL* rdzdrd0. Then I = 2па 4па па None of these

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Q: Use the contraposition to prove:  If n^2 + n + 1 is an even integer, then n is an odd integer

A: Step 1:  Step 2: Step 3: Step 4:

Q: Prove a^2 +4a+3 is odd if and only if a is even.

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Q: Let p, q, and r be prime natural numbers with p # q and n= p2q. Then, all the subfields of GF(r")…

A: Answer: Option (C) Given that, p,qand r are prime natural numbers and n is positive integer. Let F…

Q: Prove by cases For any integer n, n^2 + 5 is not divisible

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Q: Show that is composite for all primes p. p+14p²-8

A: The composite primes are those numbers which are generated by multiplying the two smallest positive…

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 or disprove: There exists an integer a such that 14 | (3a – 5) and 21 | (2a +3).

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Q: 1. Show that 2465 is a Carmichael number.

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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: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: - 5. Prove that an integer n is odd if and only if n° – 3 is even.

A: Answer

Q: 2 We know that & is clased under "+" and Moreover,we know that ヒnaw +hut B ER-8 and (5 ER-Q. Prove…

A: Given that3∈R-Q and 5∈R-Q Have to prove 3+5∈R is not a rational number.

Q: 5. Prove that n² is an even number if and only if 3n + 1 is an odd number by proving a) If n² is…

A: To Prove: We have to prove that n2 is an even number if and only if 3n+1 is an odd number.

Q: Find oll integers Such thot 24! x =1 Cmod 2) 5x = 3. (mod 8) and 1500 < X <2000

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Q: For any integers a and b, a + b ≥ 15 implies that a ≥ 8 or b ≥ 8.

A: Let a+b≥15, where a,b are integers.   Case 1: b&lt;8 Then: b&lt;8⇒-b&gt;-8⇒15-b&gt;15-8⇒a≥15-b&gt;7…

Q: Consider the following theorem: There do not exist three consecutive odd integers a, b and c such…

A: In the last , I have written the theorem " There does not exist three consecutive odd integers…

Q: (m")(2*)(m)(m³) (m')(2) Simplify and then evaluate for m = 3 and n= 7. %3D O a) (m*)(2?) = 3969 %3D…

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Q: 1. Show that 2465 is a Carmichael number.

A: Let n=2465

Q: Let S be the set of numbers of the form { ((-1)^(n + 1)) (n - 1)/(n + 1) } where n is an element of…

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Q: is there integer that has reminder of 2 when its devided by 5?

A: There are infinite number of integers that has reminder of 2 when its divided by 5

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: If p, q, r are prime numbers such that pqr = 19(p + q + r) then the value of 2(p + q + r) is

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Q: Prove {12a + 4b : a and b are integers} = {4c : c is an integer}

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Q: Ifnzo integer with (n+1) (n+3) is ood, then (n+2) (n +9) must be a multiple of 1936 4) None

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